To rely is to have confidence in something. You might rely on staying in bed to make you feel better and rely on your friends to tell you everything you missed when you were sick.
Rely is a lot like the word depend. If you rely on someone for help, you also depend on her. But rely is even more powerful! When you rely on someone, you have total faith and trust that she will come through. It comes from the Latin word religare which means "to fasten or bind." You're bound to the people you rely on!