Russian covered the mechanics pretty well.
99% of the time you see triggers used with acoustic drums they're to replace mics for a controlled sound. In a lot of cases (sadly) the drum heads are muffled quite a lot, so all you hear is the triggered sound. There's nothing wrong with that, I just think that if that's what you want, play electronic drums.
On the other hand, lots of people have used a combination of mics and triggers. Usually live, but I've seen it in the studio as well. This is done especially with the kick and snare. As with a lot of things, you can get pretty creative doing this. Usually, though, it's just to reinforce the live sound, especially to beef up the bass drum.
The basic process for using triggers (or pads for that matter) is that each one will plug into trigger inputs on the back of the drum module (sometimes called a drum brain). From there you can use the pre-programmed "kits" (also called banks), or you can reprogram those to suit your needs. When I say "program" that doesn't mean writing code or anything. It just means selecting the sounds you want, tweaking those sounds, maybe dialing in some effects or something. Some systems can get really complicated (like Roland's V-Drums), but most of them are pretty straight-forward to use.
Hope this helps.