The pronoun "their" is the possessive form of "they." It is used as a modifier before a noun. For example, they were excited to move into their home, and citizens do not always know their rights.
"They're" is the contraction of "they are." For example, they're going to get ice cream. It has the same meaning as they are going to get ice cream. It would be incorrect to say their going to get ice cream.
The adverb "there" means a location or point. For example, over there is a grocery store. The pronoun "there" introduces a clause or sentence, as in there must be another exit. The noun "there" means a point or condition. For example, we went on from there. There is also an adjective to add emphasis, as in ask that man there.
The interjection "there" is used to express feelings such as relief, satisfaction, sympathy or anger. One example is There! Now it's done!