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.

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