Finiteness (/ˈfaɪnətnəs/) is a grammatical property of verbs; verbs with a subject are finite (/ˈfaɪnaɪt/), whereas those without a subject are non-finite (/ˈɪnfənət/). Similarly, finiteness applies to clauses; finite clauses and non-finite clauses.
Finite verbs have a subject, that may either be an expressed subject or an implied subject. In English, finite verbs are marked for tense, and may mark grammatical person (albeit only in the third person singular, indicative mood, i.e. with the third person -s).
Non-finite verbs are verbals, i.e. infinitives and participles. There are two infinitives in English; the bare infinitive and the to-infinitive. There are two participles; the past participle and the present participle. Participles are marked for aspect.
Do not confuse with definiteness.