Frequently confused, the distinction is as follows:
They're is the contraction of they are.
- They're having lunch at the Staff Club today.
Their is the possessive pronoun and can be used instead of his or her.
- Their desk is very messy.
There is when you are referring to a place or to state the existence of something.
- Where do you live? Over there by Poppa's Pizza.
- There is an OUSA event on in the Octagon today.