Finite verbs are those verbs in a sentence that take a subject and have a tense. They are not a category of verb. Instead they refer to the form that the verb is taking.
So all verbs can be finite, it just depends what function they have in the sentence. Take for example the verb 'to go'. In the first case it is finite, but in the second case it is non-finite:
The first 'go' has a subject (I) and can express a person and tense (e.g. goes, went) but the second one has neither of these. It has no subject (it is instead just saying more about the verb 'want') and you cannot say 'I wanted to went...', so it cannot express tense.
We will learn more about finite verbs and their differences to non-finite verbs below.
To understand them better, here are some examples of finite verbs in sentences. Note they all have a subject (underlined) and could be used in the past, present or future tense:
There can of course be more than one finite verb in a sentence, as long as it is still connected to the subject, or a new subject, and it can express a tense:
Although a clause with a finite verb must have a subject, it would still be a finite verb if the subject were omitted for some reason, such as when it is implied.
For example in this case the subject is the same for both verbs even though it is appearing in front of only the first one:
Or in commands which often have no subject stated:
And also in utterances such as these:
In contrast to finite verbs, non-finite verbs do not have a subject nor do they express a tense.
So that means that while finite verbs can act as the root of a full sentence or an independent clause, non-finite verbs cannot.
For instance, look at this sentence:
'Walked' is finite because it agrees with the subject (boy) and expresses a tense (past simple). However, the verb 'buy' does not agree with the subject or express a tense.
'buy' is in fact an infinitive. Other non-finite types of verbs are participles and gerunds.
These are found in the perfect and continuous (progressive) tenses:
Also for making adjectives:
When a verb functions as a noun it is a gerund. Again in all these cases you'll see it is not taking a subject (it is the subject in example one) or taking a tense:
One area of confusion for students of English is whether modal verbs (e.g. will, could, may, might, should, must etc) are finite or non-finite.
It would make sense that they are non-finite verbs because they do not conjugate for tense or have a third person singular (i.e. taking an 's' after he, she, or it).
But they are followed by the infinitive, so this second verb cannot be finite either, which will not make sense because every clause in a sentence must have a finite verb!
In fact modal verbs are an exception to the rules above. They are finite even though most do not express tense. A finite verb will be the first one in a string of verbs, and the modal verb is always the first one, coming before the main verb.
So in these cases, the modal verb is the finite verb in the sentence (the verbs following the modal which are non-finite are underlined):
So modal verbs are given the status of finite verbs even though they do not express tense or person like other main verbs.
There is a good discussion here on whether modal verbs are finite or non-finite.
As all verbs can be finite or non-finite it may seem odd to study them and so they are often not learned by students of English.
It can be useful to refer to them though because it can help students to understand that only one verb in a clause (not sentence as a sentence can have several clauses) will express a tense as sometimes students will attempt to add a tense to verbs that have no subject.