AF-S is AF 'single'- your camera will focus on a fixed object when you press the AF button and will remain focused on that point for as long as you hold the button down. This is best for static subjects.
AF-C is AF 'continuous' - the camera will focus on whatever is in the relevant AF points (depending on how you have your AF coverage set up) and will attempt to keep the subject in focus wherever it moves, whether that's towards/away from the camera or laterally. This is best for continually-moving subjects, e.g. birds in flight.
AF-A is AF 'auto' and automatically switches between the two AF modes depending on what the subject is doing. This requires less effort on the part of the photographer, but you are relying on the camera guessing correctly what you are focussing on, and may result in unexpected focus.
For completion, on a Canon camera the equivalent modes are, respectively, One-Shot AF, AI Servo AF and AI Focus AF.