Now here’s another impossible task: telling the difference between there, they’re and their. I’m saying this because so many people seem to get it wrong, not because it’s actually difficult. It’s quite the contrary as we shall soon see.

You should use there when you are referring to a place as there is an adverb that substitutes a location.
Example: We were there last night.
As a general rule: if you replace there with here and the sentence still makes sense, then you have chosen the correct form.
Example: We were here last night.

There is also used with the verb to be (in the form of is, are etc.) to specify that something exists.
Example: There are 9 million bicycles in Beijing.

Their is a pronoun that indicates possession (something that belongs to them, if you will).
Example: This is their house.
To test the usage, you can replace their with another possessive pronoun (like our). As before, if the sentence still makes sense, then their is the correct form.
Example: This is our house.

Finally, they’re is made out of two words: the pronoun they and the verb to be (in the form are). Needless to say (but I will say it anyway), if you simply replace they’re with they are and the sentence still makes sense, then you have chose the right form.
Example: They’re trying to study. / They are trying to study.