THERE

There is an adverb. It refers to a special place and means in (or at or to) that place.

I have read a lot about Stockholm and I have always wanted to go there.
You can put you bag over there.
We go there every summer.

There is also used in a more abstract sense:

Hello there!
There you go (=Now it starts again).
There, there, it won’t hurt much longer now (said to comfort).

There can also be used as a formal subject, usually to say that something exists.

There is a nice pub just round the corner.
There is a new edition of her guidebook.
There seems to be a slight mistake here.

While there has many functions, their and they’re have only one meaning each.

THEIR

Their means belonging to them.

Have you seen their new house?
Their daughter is fifteen.

THEY’RE

They’re is a contracted form of they are.

They’re waiting for us at their hotel.

See also my post about it’s and its.

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