A finite clause is one with a tensed verb.
He woke up tensed early.
A non-finite clause is one with an infinitive or participle, which are tenseless, and therefore not on a time-line. Non-finite clauses often serve as a nominal grammatical subject.
To wake up infinitival after a night of carousing can be difficult.
Waking up participial after a night of carousing can be difficult.