Adagio's Small Guide to Understanding the Difference Between There, Their, and They're

There - adverb/preposition - in, at, or near a certain point; also used along with or as an exclamatory interjection.
Not to be used in reference to any number of persons' ownership of something or a group's current condition, psychological or physical.
E.g.: "There, upon the hill, lies a small tree."
"Somebody is over there."
"There! Look!"

Their - pronoun - used to express ownership of something by a group excluding the narrator. Often used informally in the place of his/her and the more formal "his or her" in the context of something or someone whose gender is not known.
Not to be used in reference to someone's or something's current position or any number of people's current condition, et cetera.
Not to be used in formal speech, documentation, writing, et cetera in the place of "his", "her", or "his or her".
E.g.: "Their form is incredible."

They're - a contraction of the unisex pronoun they and the present tense form of the verb "be", are. Used to express a group's current collective condition. Similarly to "their", it is often used informally in the place of he/she is and "he or she is" in the context of unknown gender.
Not to be used in reference to someone's or something's current position or any number of persons' ownership of something.
Not to be used in formal speech, documentation, writing, et cetera in any given circumstance.
E.g.: "They're coming through the valley."