The word "they're" is a contraction of "they" and "are". It seems people have forgotten what contractions are, and what the apostrophe means. In all contractions, the apostrophe is standing in for missing letters. My first grade teacher taught us about contractions this way, and it has helped me keep them straight ever since. (I'm not telling you how long ago that was, however!)
Take the word "they" and the word "are". Smash them together. They spell theyare. Keep smushing. The letter "a" gets wadded up and becomes the apostrophe. "They're".
The word "their" is a possessive pronoun. If something is "theirs" it belongs to them. Remember it this way: It is "theirs", it belongs to "the heirs". "Theirs" contains the word "heirs". (This one isn't actually a contraction, though; it's just an easy way to remember it.)
The word "there" is a location. The object is over there. It coincidentally contains the word "here". If you need a location, the thing you're looking for is either "here" or "there".