help w/ Orbits and Stabilizers

So in my abstract algebra class today we went over Orbits and Stabilizers, but my teacher did a horrible job of explaining it, and the book isn't much help either, so I was wondering if anyone had any nice online tutorials about orbits & stabiliziers, or if someone here could just explain the general concept and what the "point" of them are. We also discussed group actions and that didn't seem to make much sense easier. I guess I understand what orbits and stabilizers are, but I'm failing to see how they fit into the bigger picture of abstract algebra.