What do you have to concider when putting a new stabilizer on a bow. how do you know if you have the right one? Also what do they call the thing that stops your string from going forward when you shoot.

You want one that provides enough weight that allows you to steady your bow without feeling too heavy to you.I know that's a vague answer. However, it seems that most do the same thing and it comes down to what looks best to you.

You are talking about a string suppressor. There are STS brands as well as ones made by MeanV (he's a member here) and at least 1 other manufacturer that I've seen in stores.

My stabilizer is more for vibration/sound dampening than stability. I think my General would shoot just fine without a stabilizer.