Their is a possessive pronoun. It always describes a noun.
Note the spelling of their. It comes from the word they, so the e comes before the i.
There is an adverb meaning "that location." It is sometimes used with the verb to be as an idiom. It is spelled like here which means "this location."
They're is a contraction of they are. Note the spelling: The a from are is replaced by an apostrophe.
Examples: Their dog has fleas. (possessive of they)
I put the collar right there. (that location)
There are four prime numbers less than ten.
(with to be)
They're 2, 3, 5, and 7. (contraction of they are)
See also There's, Theirs, and Their's, Possessive Pronouns and Verb Contractions.