How many different groups of dinosaurs are there?

It depends on what size "group" one is talking about. There were two big groups, the Ornithischians, whose members had hips like those of modern birds, and the Saurischians, whose hips were more lived those of lizards.

Within these two groups, there were smaller, but still pretty populous, groups of Theropods, Sauromorphopods, Stegosaurs, Ankylosaurs, Ceratopsians, Hadrosaurs, Pachycephosaurs, Dromaeosaurs, Ornithomimi, and probably others that haven't been discovered.

The smaller groups are the Genera, and these are the names you know: Apatosaurus, Tyrannosaurus, Triceratops, etc.