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.