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.

What exactly do you mean "how it fits into the bigger picture"? How the concept of orbits is applied? A pretty classic proof that a permutation cannot be both odd and even relies heavily on orbits? Also, the definition of a cycle (in most cases) is a permutation that has at most one orbit containing more than one element. The length of a cycle is the number of elements in it's largest orbit.

Any decent book in algebra (Dummit and Foote, Hungerford, Lang, Hershtein, etc.) covers this topic, which is one of the most important and far-reaching in group theory, with many branches in many other parts of modern mathematics, so you better get a good grasp of it now, that you're beginning to study it.

Tonio