Both forms are correct. Their and they’re are two correct forms, but with different functions in the sentence. Both forms can be easily mistaken because they are homophones, which means that they sound exactly the same despite different spellings. Their is a possessive form of the pronoun they and determines a sense of belonging to them. Meanwhile, they’re is a contraction of they are and expresses a state of being. To sum up, besides the same pronunciation these words don’t have much in common and are used in a different ways.

Correct spelling, explanation: it’s a determiner of a personal pronoun they, which expresses the possession or belonging of something.

Correct spelling, explanation: it’s a contraction of a phrase they are, a present plural form of a verb to be.

Definition of their:
pronoun: belonging to them; can be used to avoid his or her;
John and Susan asked me to take care of their dog during their vacation next week.
Someone forgot their wallet, should we check if there are any documents inside?

Definition of they’re:
verb: contraction of you are; to live, happen, occupy a position;
They’re visiting my parents this weekend and will drop in on their way back.
They’re too irresponsible to take over the family business.