Its is the possessive pronoun; it modifies a noun.
It's is a contraction of it is or it has.
Incorrect: The mother cat carried it's kitten in it's mouth.
(Possessive pronoun, no apostrophe)
Correct: The mother cat carried its kitten in its mouth.
Correct: I think it's going to rain today.
(Contraction of it is)
Correct: It's been a very long time.
(Contraction of it has)
See also Apostrophes with Pronouns.