Their is a possessive adjective. It always precedes a noun and indicates possession:
Is that collie their dog?
That cabin is their country residence.
They’re is a contraction of the words “they are”:
They’re interested in buying your house.
Do you know if they’re at home?
There has more than one function. It is used as an adverb of place:
See that Victorian house? I lived there when I was nine.
I think I left my keys in the kitchen. Please look to see if they are there.
There is used as a sentence opener:
There is a tide in the affairs of men,/Which, taken at the flood…
Yes, Virginia, there is a Santa Claus.
Note: Beginning a sentence with there is sometimes the correct stylistic choice, but a common error is to begin a sentence with there when beginning with a subject would be better. For example:
Weak: There will be a brass band to meet the war hero at the airport.
Better: A brass band will meet the war hero at the airport.
Weak: There are plenty of reasons for her to refuse his advice.
Better: She has plenty of reasons to refuse his advice.