There: a place
There refers to a location or is used to introduce a sentence ("there is/are"): "Put the box over there." "There are three options."
Their: possession
Their shows that something belongs to a group of people: "They forgot their umbrellas." "It's their decision."
They're: contraction of "they are"
They're is short for "they are": "They're coming to dinner." "They're not ready yet."
The trick
Try replacing the word with "they are" — if it fits, use "they're." If the sentence is about ownership, use "their." Otherwise (location or "there is/are"), use "there."
Frequently asked questions
How do I remember there vs. their vs. they're?
Test with "they are": if it fits, use "they're." If it's about ownership, use "their." Otherwise, use "there" (place).
"Their house is over there" — is that correct?
Yes: "their" (possession, the house belongs to them) and "there" (location) are both used correctly.
Does "there" have another common use besides location?
Yes, to introduce a sentence: "There is a problem," "there are many reasons."
Why are these three words so often confused?
Because they are homophones — they sound exactly the same in spoken English, so the difference only shows up in writing.