Their, they're, there which is correct

All of them, but it depends what you are saying to make it correct.

Their is the possesive pronoun for they, They are happy because their car is fixed now.

They're is the contraction form of the afirmative form of the third person plural of the verb to be. They are = They're.

There means a place away from oneself. Also it forms along the verb to be the expression there is / are to denote existance of something.

There is a car outside. (singular)

There are two windows in my classroom. (plural).