Mountain bike rear derailleur?

I am kinda new to the whole mountain bike scene. I was wanting to get a new rear derailleur I am stumped by how many different one there are. I have a mongoose and I know it has 8 gears on the the rear wheel and 15 teeth and 13 teeth on the rear derailleur gears. Now do I have to get the same teeth and what is the difference on a short and long cage?