Theirs is a possessive pronoun.
There's is a contraction for there is or, rarely, there has. Note the apostrophe replacing the missing letter or letters.
Their's does not exist.
Examples: That painting is theirs.
There's more to this than meets the eye.
(contraction of there is)
See also Possessive Pronouns, Verb Contractions, and There, Their, or They're.