What do "finite" and "non-finite" mean in linguistics? I know that they occur in other languages and in some cases not only in verbs.
In traditional grammar a finite form of a verb is a fully specified verb form according to all verbal categories relevant to the specific language, like voice, aspect, mood, tense, person, or number.
Non-finite verb forms are underspecified in this respect, leaving out some of the categories required for a finite verb form (typically tense, person and number). Examples of non-finite verb form include infinitives, gerunds, participles, or the Latin gerundive. Languages vary in the number of non-finite verb forms they have, and there are languages not having them at all.