Categorizing roleplaying games is difficult, I'll leave that to those technically minded. But if I'm allowed I'll split roleplaying games in two broad categories it would be the destinction between on-line and off-line games. The first is done by connecting players to a computer program somehow. The players type in commands to tell their character what to do, and they see what the other characters in the same location do and say. The second is done usually with a board or card-like game. Players come together to play a game that involves a fair amount of rolepaying.
The on-line version has the advantage that you are very much involved first person. It's awfully easy to ignore that you really are sitting behind a keyboard and typing a conversation. The off-line games have the advantage that you are actually meeting people. Often that adds a certain flavour to games, and it alows you to have fun together outside the game as well as inside it.
Actually now I come to think of it there's a third category (which proves that I really shouldn't bother with all this definition stuff): live-acting roleplaying. Here a group, and sometimes it's a huge group, is brought together and they're set a task or quest. The difference is that this time they -are- the character. No typing, no boards and dice, but a real forest or something and real actors to play the opponents. You don't hear or read stories of what is happening, they happen to you there and then. If you are to fight you are to fight for real (or rather as real as can be without actually hurting the other). Games like these are awesome but unfortunately they're also quite difficult to organize, or expensive to take part in. If you ever get a chance though, take it!
A less expensive, and less game-like, form of live-acting is what is commonly known as living history. Here a group of people re-enact a historical period or event, e.g. life in the 17th century frontier villages or the battle of Waterloo. This too is great fun and most organisations are happy to welcome new members.