The division of clauses into finite and non-finite is fairly recent. Until a few decades ago, clauses with grammatical subjects and finite verbs were known in traditional grammar as clauses, while those with non-finite verbs were called phrases--infinitive phrases, participial phrases, etc. This nomenclature has been largely superseded by the classification "finite/non-finite."

A finite clause has, at the very least,

1) a grammatical subject (except for imperatives like "Listen to this," which have an understood second-person subject)

2) a verb (the first or the only verb) marked for tense and often for number and person

The verb is a full verb, "anchored" in time.

A finite clause can be an independent clause, as in

Marvin owns a posh heath club

or a subordinate clause, as in

She stands for several principles that I don't agree with

When you decide to move to the new apartment, you can stay with me until you're settled

A non-finite clause does not need a grammatical subject. The verb is incomplete, that is, it does not carry markings for tense, number, or person.

The verb can be 1) the infinitive:

(from Google) Gary`s greatest ambition is to walk backwards without actually moving, like the singer Michael Jackson.

2) the past participle :

Stunned by his criticism, she quit her job and moved to Tibet

or 3) the present participle:

Melinda, raising her voice over the crowd, called for order

In short, a finite clause has a full verb, while a non-finite clause has a verb form that is not marked for tense.