Finite and non-finite verb forms

Definitions

The first thing to define is the term verb. In what follows, verb generally means verb phrase.
A verb phrase may be:

a single word: as in, e.g.:
I went to London
a head verb with an auxiliary verb or verbs before it: as in, e.g.:
I could have gone to London.
a verb or verb phrase modified by an adverbial: The adverbial can be an adverb, a noun phrase, a prepositional phrase or another clause, as in, e.g.:
    I eventually managed it (adverb pre-modifier)
    I have finished this morning (noun phrase post-modifier)
    I studied at university in the 60s (prepositional phrases post-modifiers)
    I was happier when I worked alone (finite clause post-modifier)
This is not an analysis universally accepted. In some analyses, a verb phrase can only consist of verb forms, whether finite or not. For teaching purposes, because adverbials are so often embedded, it makes some sense to treat the whole item as a single unit.

What's the difference between finite and non-finite verb phrases?

There are some distinguishing features of the two types of verb phrase:

Finite verb forms

Finite verbs and verb phrases are clearly linked to definable subject and exist either in the present or past tenses. They show:

Tense
For example, by changing the ending or the central vowel as in:
    come to came
    teach to taught
or
    arrive to arrived
Person
For example
by adding an -s for the 3rd person singular:
    smoke to smokes
or
    go to goes
or making other changes to the verb form as in:
    be to am, is or are
    have to has

English is, in general, not a highly inflected language so many finite forms in English are not marked for person or tense. They are, however, still finite forms although the form is identical to the base form of the verb. For example, in:
I go
the verb form is identical to the infinitive (which is unmarked).

In the trade, this is known as zero marking (usually signalled by Ø) which is not the same thing as no marking at all. Many items in the language are zero marked such as:

articles as in, e.g.:
Take Ø tools with you
plurals as in e.g.:
There were lots of deer(Ø) in the forest
relative pronoun clauses
She's the one Ø I gave the money to

In all these cases, the item is absent but still performing a grammatical function.
Non-finite clauses are, however, different because they are not marked for person or tense even by zero.

In more obviously inflecting languages such as German, Spanish or French, the verb would be marked both for person and tense.
In this case, when discussing English, we can speak of a zero inflexion (Ø inflexion).
The translation is, for example:
    In French: je vais
    In Spanish: yo voy
    In German : ich gehe
    In Italian: io vado
    In Czech: já jdu
and in those languages, the verb form is marked for first person singular.
We do, as we saw above, mark the verb form for tense in English changing, for example:
    I like
to
   I liked
with the addition of the final 'd' but this remains the case for all other persons, too, so only the tense is marked. In other languages, the past ending varies from person to person and is, therefore, doubly marked.

Incidentally, one of the reasons that pronouns may not be omitted in clauses in English is precisely because the reader or hearer needs the pronoun to make sense of the verb so, for example:
    came
carries only the meaning of the verb but does not signal who came.
Other languages, referred to as pro-drop languages, have no need of the pronoun because the verb form itself signals the person so, for example, in Italian:
    I came
may simply be encoded as:
    venuto.

Central modal auxiliary verbs do not have marking either for tense or person. There is, then, no -s marking on such verbs for the third-person singular and, in the case of many verbs, the past tense employs a different verb altogether so the past of must, for example, is had to and so on. Central modal auxiliary verbs are, nevertheless, always finite, never non-finite.

Non-finite verb forms

Non-finite verbs and verb phrases are not specifically tied to a subject and do not show tense or person. They can be:

gerunds (i.e., verbs which behave grammatically in a noun-like way). For example:
I enjoy walking
They are not actually just simple nouns in most cases because they cannot take plurals and are treated as singular forms. Some are really very noun-like however as in:
The furnishings were lovely
The distinction here is between a gerund proper, a verbal noun and a dverbal (which is a slightly fuzzy area, in truth).
1. Gerunds retain verb-like properties and cannot usually be modified by adjectives as nouns can, they may take direct objects as verbs do, they are modified by adverbs, just like verbs, they form singular mass noun concepts and they are only formed by adding the -ing suffix. The following contain gerunds, therefore:
      My saying that was a mistake
      Fishing is idiotic
      Walking slowly in the country is a pleasure
  Gerunds are non-finite verb forms.
2. Verbal nouns, on the other hand, are nouns derived from verbs and are more noun like but not fully so. They can be modified by adjectives rather than by adverbs, do not take direct objects and are almost always uncountable (mass) concepts and cannot be made plural.
  The following contain examples of verbal nouns:
  Loud shouting makes her nervous
  Angry arguing could be heard all over the house
  Long-distance running is my hobby
  Verbal nouns are not, therefore, non-finite verb forms, they are nouns.
3. Deverbal nouns are nouns derived from verbs which are fully noun-like in their behaviour and syntactic properties. They can be made plural and can be formed from verbs in ways other than with the -ing suffix.
  For example:
  The furnishings were very expensive
  The new buildings are complete
  Her flat refusal upset everyone
infinitives of two sorts:
1. bare infinitives as in, e.g.:
  Let her come
  She might object
2. to-infinitives as in, e.g.:
  I want to help
  She used to be so happy
participles of two sorts;
1. present participles as in, e.g.:
  On arriving, I saw the party had nearly finished
  Smiling, she welcomed him in
2. past participles as in, e.g.:
  Broken, the hammer was no use
  The aria was beautifully sung

Central modal auxiliary verbs are, again, an exception. They have no non-finite forms. We cannot form a participle of either kind from, for example, may, might, will, could, must etc. and nor can we form an infinitive or gerund from such verbs.
Central modal auxiliary verbs are, therefore, always finite but formed without inflexion for person or (usually) tense.
The other, primary, auxiliary verbs, do, be, have, get are not restricted in this way and routinely form parts of non-finite verb phrases.

Non-finite verb functions

Non-finite verb forms function grammatically in five distinct ways:

Adjectives
For example:
It's a smoking gun
The window's broken
Nouns
For example:
I like playing cards in the evening
I hate waiting for buses
Infinitives (with and without to)
For example:
    I came to help
    I want to complain
    She must help us
Present participles
For example:
Falling over, John hurt his ankle
I was living in London at that time
Past participles
For example:
Disappointed, he left early
The car was repaired

Both finite and non-finite forms may occur separately or in the same verb phrases and both may be modified by adverbs and determiners. For example:

John went to see about the problem.
I very much enjoy sitting doing nothing in the garden.
They quickly spent all their money buying junk.
I really am not near finishing the work.
She came hoping to talk to her sister
They would watch TV for hours on end

As a short exercise, comment on the verb phrases in bold in the following, deciding if they are finite or non-finite.
Then click on the to reveal some comments.

She has stolen the money	This is a finite clause identifiable from the inflected form of the verb. It contains the non-finite -en form, stolen and the verb have is marked for person with the form has.
It’s tidy	This is a finite clause with an inflected form of the verb be.
Smoking is banned here	This is a non-finite verb operating as a noun. In this case it's a gerund and uncountable.
I'm smoking too much	This is the same form of the verb but in a finite verb phrase with an inflected form of the verb be. It contains the non-finite smoking.
It made me sick	A finite verb phrase identifiable by the inflected past form of the verb make.
You must go now	A finite modal auxiliary verb with the non-finite bare infinitive.

In the third example, there's a finite form in the passive (is banned) and in the last example, there's a non-finite form: go (that's what the term infinitive implies, of course).

Confusions

Mostly because English is weakly inflected, a good deal of confusion can arise if we are not alert to the existence of zero inflexions.

Some confusion is caused in this area by the fact that verbs in English do not show very many inflexions. A verb may look the same but be performing different grammatical functions. In, e.g.
    I am playing
we have a finite verb phrase (with the non-finite -ing participle) but in
    I enjoy playing
we have a non-finite use of the verb play and a finite use of the verb enjoy which, in English, is not marked for person except in the third person singular when an 's' is added.
Equally, in
    The game is played here
we have a passive finite verb phrase with the marked form of the verb be and the non-finite -ed / -en participle and in
    A much played game
we have a non-finite use of played as an adjective modified by the adverb much.

It is possible to describe individual verbs (rather than verb phrases) in the same terms. For example, the verb playing in
I am playing
is often described as a non-finite form (because it carries no marker for tense and person).
However, in combination with the verb be, as in He is playing, the form is part of a finite verb phrase. We need to be careful with our terms and distinguish between a finite or non-finite clause and a finite or non-finite verb form.

The -ing form and the -ed / -en forms are both non-finite and often called present and past participles respectively. This is slightly misleading insofar as:

Regular verbs take -d or -ed depending on their morphology in both the past tense and the perfect forms so it becomes impossible simply by looking at the verb to say whether, e.g., played is the finite past tense of play or the non-finite past participle in, e.g., They have played.
The same confusion can arise with irregular verbs, as many, such as catch, teach, read etc. have the same form for both past tenses and non-finite participles. Some verbs, such as put and set exhibit no changes at all for tense.
For this reason, many analyses distinguish them by calling the past tense form the -ed form and the past participle form the -ed / -en form. That is the system used here.
The -ed / -en form (or past participle) is not confined to past tense structures. It occurs for example, in
She will be invited
where it forms part of future passive form and
She is having her house redecorated
where it forms part of a present-tense causative structure.
Calling it the past participle may lead learners to conclude that it can only be used in past-tense structures.
The so-called present participle or -ing form exhibits similar problems because it can appear in non-present tense forms such as
She had been running to catch the bus
or
We will be spending our holidays in France next year
Again, calling it the present participle can mislead.
The -ing non-finite form has multiple functions. It can, for example,
1. form a gerund in:
  She dislikes swimming
2. form a verbal or deverbal noun in (which are, as we noted, nouns not verbs strictly speaking):
  The abstract paintings were sold
3. form part of a progressive tense form in
  She is swimming
4. form part of an iterative (repeated) aspect form in:
  She was singing in the choir during her teenage years
5. form part of a continuous aspect form as in
  She was hoping for a pay rise
6. be an adjective as in
  She handed over the smoking gun
The -ing form is often said to be either a participle or a gerund but that, too, leads to confusion. The form can, in fact, occupy an intermediate position between a gerund proper (or a verbal / deverbal noun) and a participle (i.e., part of a finite clause). In, for example:
    The buildings were damaged
the noun buildings has been derived from the verb build but functions only as a noun (taking a plural form and not allowing an object as the verb does). Other derived forms work similarly, such as booking, carving, christening, clipping, covering, crossing, drawing, failing, flavouring, heading, meeting, mooring, offering, painting, peeling, rambling, ruling, saving, setting, shaving, sighting, swelling, turning, warning etc. which can all be made plural and rarely take objects. Whether these should be described as gerunds is doubtful and on this site we prefer the terms verbal or deverbal noun to describe such formations.
However, in
    Playing the bagpipes is difficult
the word playing is slightly more verb-like insofar as it cannot take a plural (as countable nouns do) but may be treated as a mass noun although it clearly takes an object (the bagpipes) which nouns cannot do.
It may also appear with a possessive determiner as in:
    His playing was awful
where it is noun-like insofar as it forms the subject of the copular verb (was) with the complement attribute, awful.
Compare, for example:
    His music was awful
which is parallel but clearly has a noun as the subject.
The form can, of course be fully verb-like in, e.g.:
    He was playing the bagpipes
when it forms part of a finite transitive verb phrase.
In between, we encounter some less obvious cases of noun-like and verb-like behaviours mixed and where the line is drawn is difficult to see. For example, in:
    I heard him playing the bagpipes
the reference is clearly to a progressive event (i.e., a finite tense form proper) and the -ing form takes an object. It is somewhere in between and could be classified as a gerund or a participle.

Verbs of perception

These verbs, see, hear, notice, observe, smell, sense, watch, feel work slightly differently with non-finite forms.
The key is to whether reference is to the whole action or a part of it, usually. For example:
    I saw him drink the beer
implies that I saw the whole action, from the full to the empty glass, whereas:
    I saw him drinking the beer
implies that I only saw a part of the action which started before and finished after my observation.
Equally, there is a difference between:
    I heard her sing at the concert
and
    I heard her singing upstairs
in which the first implies that I watched the whole performance and the second that I heard only part of her singing.
Compare, too:
    I saw him sit down vs. I saw him sitting in the restaurant
    I watched her repair the car vs. I watched her repairing the car
    I noticed him talking to the manager vs. I noticed him talk to the manager
    I felt the dog touch me vs. I felt the dog touching me
    I smelt the toast burn vs. I smelt the toast burning
and so on.
The difference stems from the sense of a progressive or background aspect with -ing forms of verbs in English and completed action with bare infinitives.

Many languages, which do not mark progressive forms will not distinguish in this way and speakers of those languages need a certain amount of help to see the difference. For example, the English sentences:
    I saw him drink
and
    I saw him drinking
will both translate identically into German, Spanish, French, Italian, Dutch and Swedish and many other languages. Respectively, the translations would be:
    Ich sah ihn trinken
    Lo vi beber
    Je l'ai vu boire
    L'ho visto bere
    Ik zag hem drinken
    Jag såg honom dricka

Finite verb forms

Finite verbs in English are sometimes identifiable by the changes to the verb form. Figure out which of the following English can show by changing the verb form and then click on the to reveal some comments.

Gender	English has no way to show this either in the singular or the plural. In some languages, the verb form will change depending on the gender of the subject (or the speaker). Hindi is a major example.
Person	The -s inflexion in the present simple of a verb indicates the third person singular. This is the only person inflexion on regular verbs in English.
Number	The verb be distinguishes plural from singular (am vs. are, is vs. are) but the system is incomplete and most verbs only use the third-person -s to note singular vs. plural (he/she/it goes vs. they go, we go, you go, I go etc.).
Tense	This is indicated for most verbs by a change of form or the addition of a suffix (smoke-smoked, come-came etc.). Some common verbs, such as put and set, do not even show this change for tense. All regular verbs, as we saw, use the same -ed or -d ending for both the participle and the past tense and many irregular verbs also use the same form for both.
Aspect	English uses auxiliaries to indicate aspect (the verb be for the progressive [she is arriving now] and the verb have for the perfect [she has arrived]). Many other languages, such as German, arguably do not mark aspect at all.
Mood	English only omits subjects in the imperative (Go home!), sometimes uses an uninflected base form to indicate the subjunctive (If it be him) but does not otherwise indicate mood except by the use of modal auxiliary verbs. In many other languages, mood is signalled by the subjunctive form of the verb.
Voice	English has only two: active and passive. The voice is indicated by the use of the verb be, get or have (the last in causative structures).

As you can see, English finite verbs are barely inflected at all. Other languages do things very differently. In some languages, all of the above may be indicated by a change in the finite verb form and most inflected languages will show a greater range than English.

Non-finite verb forms

English has only three non-finite forms:

infinitives (with and without to):
he must go
I want to help
participles (past and present):
He has left
She is running
gerunds:
He dreads meeting her
Overeating is a cause of illness

The distinction between a participle and a gerund is by no means as clear cut as this classification would imply as we saw above. It is probably better to consider a cline from purely participial use:
    He is meeting his wife at the airport
at one end and purely gerundial use
    That was an awful meeting
at the other, with less easily categorised forms in between
    I watched him running
and
    I objected to his running).
For more, see the article in response to a visitor's question.

However, non-finite forms appear all over the place in different guises. What are they in the following?
Decide and then click on the to reveal the answers.

*Running* is tiring	Gerund as the subject
I hate *running*	Gerund as the object
He is good at *running*	Gerund as the complement (or object) of a preposition
I want *to go*	Infinitive with to after a main verb (a catenative structure)
He let me go	Infinitive without to after a main verb (another catenative structure)
*To go* would be foolish	Infinitive with to as a subject
There's no call *to go*	Infinitive with to post modifying a noun phrase
She wanted *to go*	Infinitive acting as the direct object of the verb want
I am *going*	Present participle
I have *gone*	Past participle
I was *forbidden* *to run*	Passive participle + infinitive with to (catenative)
She was happy *to help*	Infinitive with to as an adjective complement
Once *finished*, the house looked great	Passive participle as a noun modifier

Four of these examples show something called a verb chain (they are catenative verbs to which there is a guide on this site linked in the list of related guides at the end). There are two catenating verbs in these cases, as is usual. For example,
    They agreed to try to come
    The hoped to persuade her to come
    They remembered asking me to help
where the non-finite forms have no independent existence.

Non-finite verb forms as subordinators

One important function of non-finite verb forms in English is to subordinate one clause to another. The forms can be used for many types of subordination instead of the usual subordinating conjunctions.
For example:

Temporal subordination
Instead of:
    When he opened the bonnet he saw the problem (using when as a subordinating temporal conjunction)
we can have:
    Opening the bonnet, he saw the problem
and instead of:
    After he had arrived at his hotel, he called me
we can have
    Arrived at his hotel, he called me
Conditional subordination
Instead of:
If you say that, he'll be furious (a subordinating conditional using if)
we can have
Saying that will make him furious
Concessive subordination
Instead of:
Insofar as it is allowed, I'll claim the expenses
we can have
Assuming it is allowed, I'll claim the expenses
Causal subordination
Instead of:
    He worked late in order to finish
we can have
    To finish, he worked late
and instead of
    Because the garden was covered in snow, I could see his footprints easily
we can have
    Covered in snow the garden showed his footprints easily
Cause and effect are often signalled by non-finite forms following prepositional phrases or complex subordinators (in the third example below) as in, for example:
    To do it, he worked all night
    As a result of working all night, he got it done
    He went in order to see her
and this can sometimes be achieved without the preposition:
    Working late, we got it done
    Repaired, it worked well
etc.

Learners, incidentally, do not invariably recognise the in-built subordination that many non-finite clauses contain because some languages simply cannot do that.

Some non-finite forms have become established as conditional or concessive conjunctions and occur frequently in that role.
For example:
    Providing that you pay the rent, she can't evict you
    Granted that we can find the money, I see no problem
    Supposing it rains?
    Provided only that we have time, we'll see you on Friday

Others may take on the nature of prepositions as in, for example:
Including Mary, we shall need six tickets
I see no problem regarding the timescale

There is a guide to subordination, linked below, on this site which also considers the severe restrictions on the use of finite and non-finite clauses with subordinating conjunctions.

Block language

Block language may be defined as:

A type of language that differs from canonical linguistic structures in being reduced or compacted in various ways, so as to convey a message economically. It is used especially in notices and newspaper headlines.
(Aarts, Chalker and Weiner, 2014:50)

Non-finite forms are often preferred in such items because they can communicate the idea without the need to form fully grammatical clauses. When verbs are used at all in such language items, they are frequently non-finite forms which appear often without the primary or modal auxiliary verbs.
For example, with the non-finite forms highlighted, we may encounter:
    Where to go and what to see
    Recommended by doctors
    No smoking
    Danger: flooding
    Inflation rising again
and so on.

Finite forms also occur, especially in newspaper and website headlines, but, again, the full verb phrase is frequently abbreviated with ellipsis of tense markers (or their substitution by present-tense forms). For example:
    Hamilton wins
    Prices rise steeply
    Weaver marries school friend
etc.

The clumsiness of English

English is extremely concise in some ways. For example, the -s ending on She works indicates:

person (third)
number (singular)
tense (present)
aspect (simple)
mood (indicative)
voice (active)

However, at other times the language seems clumsy and inefficient. For example,
She might have been told
contains three auxiliaries (and a non-finite form) which separately indicate:

modality (might)
perfect aspect (have)
passive voice (been)

To make matters worse, some of these auxiliaries indicate different things at different times. For example, in
They will have been working
the auxiliary been now indicates progressive aspect and not passive voice. That can be deeply confusing for learners of the language, especially those whose first languages have different ways to signal progressive forms (if they do so at all) and the passive.

In other languages, such as Greek or Russian, most or all of these can be expressed in a single verb form (as English did in the example She works).

Related guides
conjunction	for more on how clauses are connected and links to other guides to subordination and coordination
subordination	for a guide which contains consideration of how the use of non-finite forms is restricted to certain subordinators
clauses	for more on clause structures
verb types and clause structures	for a guide to the six main sentence structures in English
phrases	for a general guide to phrase structures
infinitive	for the analysis of one sort of non-finite form
catenative verbs	for a guide to important uses of non-finite forms
nominal clauses and phrases	for an analysis of the ways finite and non-finite clauses can act as noun phrases

Take a test on some of this.

Reference:
Aarts, B, Chalker, S and Weiner, E, 2014, The Oxford Dictionary of English Grammar, 2nd Edition, Oxford: Oxford University Press