When should I use their or there?

THEIR is a possessive pronoun. Examples:
Their house is the brick house on the corner.
Have you seen their dog?
We asked the children to pick up their toys.

THERE is usually an adverb. It may mean a place or it may introduce a sentence.
Examples:
I live over there.
There are thirty students in this classroom.

When should I use a apostrophy (s')?

This site has a great explanation and examples of apostrophe uses.

http://owl.english.purdue.edu/handouts/grammar/g_apost.html

There's is a contraction for there is..as in

There's a bug in my soup.

There's a right turn ahead at the orchid shop.

There's something about Mary.