Rally Obedience is a sport where the dog and handler complete a course of stations set up to test different skills while the dog is in heel position. Each station has a sign detailing the exercise the dog and handler has to perform at that time. Verbal cues and encouragement are used and it is a much less formal type of obedience to compete in.
Obedience is a more formal dog sport - where handlers and dogs complete a set of tasks as instructed by a judge with cues and limited verbals.
Obedience at some level should be something all dogs should be trained in - whether pets or performance dogs. Most puppy people start off by attending puppy pre-school and some local obedience clubs offer great obedience classes you can sign up for. Obedience helps to build a strong relationship with your dog and also sets boundaries.