MUD stands for Multi User Dungeon. (Or Multi User Dimension. Or Multi User Daemon. I've heard all three.) The most common kind of MUD is the adventure MUD, where users log into the system and travel around the MUD world, collecting items, fighting monsters, and interacting with other users.

MOO stands for MUD Object Oriented, referring to the style of the programming language in the MOO itself. Most of the MOOs that I've encountered focus more on the social aspect, travelling around the world for the express purpose of meeting other players and interacting with them.