A finite verb is a verb which has a subject and shows tense. In short, it is a verb being a verb (i.e., a doing word) as opposed to a verbal (which is another part of speech formed from a verb).
He painted the fence.(In this example, painted is a finite verb.)He varnished the painted fence.(In this example, painted is a verbal. It's actually a participle from the verb to paint, which is being used as an adjective. A verbal is another name for a non-finite verb.