What is a finite verb?

Thinking of a finite verb as the opposite of an infinitive isn't a bad way to start, actually.

The core definition of a finite verb is one that is inflected for person and tense. Typically only finite verbs can act as the main verb of a sentence. In the following examples, the italicized verbs are finite:

I wanted to go to the store.

The girls were talking to each other loudly.

A child has gone missing.

However, the bolded segments above, while part of the verb phrase, are non-finite verbs. The first is an infinitive, the second is a present participle, and the third is a past participle, these being the primary forms of non-finite verbs in English. If you remove the finite verb from all of the sentences above you no longer have a complete sentence.