An explanation of terminology

So, you ask, what is the difference between a MUD and a MOO? A MUSH and a MUX? Before I go into specifics, a quick disclaimer. I'm going to offer some generalizations based on personal experience. Almost all of what I'll be saying here is incorrect in some instances, but I think most are broadly true.

Broadly speaking, a MUSH/MUCK/MUD/whatever is a text based multi-player enviroment, divided into "rooms," and populated by players and objects, which you can interact with in a variety of ways, from speaking with them, to picking them up, to killing them. It's rather like Zork, but with a bunch of other people playing at the same time. If none of that helps, all I can suggest is that you telnet to one and look around.

MUD: MUD's (Multi-User Dungeon, Multi-User Dimension) were the systems that started it all. They grew out of a desire for multi-user "dungeon gaming," which means that most of them are very combat oriented. The typical MUD will have little or no distinction between in character and out of character interaction, little role-playing, a lot of coded commands, and a lot of fighting and levelling. There are usually three levels of playing, with normal characters, Immortals, and Wizards (sometimes apparantly also called Imps, I'm not sure why), in ascending order of power within the MUD. Applications are rarely required for characters. (Note: I have little experience with MUD's, take this with a grain of salt).
MUSH: MUSH's, aka Multi-User Shared Hallucination and a host of other acronyms, are something of an opposite from MUD's. They are almost invariably non-combat oriented, and often heavily role-playing and in-character biased. White Wolf and other similarly 'soft' table top role playing games have a lot of presence in the MUSH world, albeit usually with modifications to make them more playable online. Wizards, the powers that be on a MUSH, and Admins, their assistants, are often both system maitenance and administrators of in character interaction, though there are certainly MUSH's with a more consent oriented enviroment. MUSH's are unlikely to have coded commands to the same extent that a MUD will, relying instead on arbitration or consent to determine the effects of actions. MUSH's are popular, but have, in my opinion, a poor code base to work from, leading to somewhat clumsy commands.
MUX: Largely similar to MUSH's, MUX's (no acronym that I'm aware of), have a slightly larger native code base, but seem to have similar biases in terms of actual play.
MOO: MOO's are object oriented, and have a C++-like internal code. Most MOO's are educational or social, but you'll occaisionally see a role-playing one.
MUCK: MUCK's (no acronym that I know of), are something of the MU*'s of choice for those who want something between the entirely combat driven MUD's and the extremely character driven MUSH's/MUX's. Several popular Anime oriented systems are MUCK's, and in these, in character or out of character interaction is equally permissable, and consent oriented game-play alternates between social interaction. In terms of code, the native base of MUCK's is extremely small, making commands which would be primitives on other systems coded commands on MUCK's. This can be both very nice, allowing full featured commands that MUSH's, for example, do not have, and very irritating, as each command takes its own protocol for help, switches, etc.

Coded commands: By this, I mean commands not native to the MU*'s native code, but which the average player does not have to do any coding for -- they are created by the Wizards/Imps/Immortals.

MU*'s: A generic term for MUD's/MUSH's/MUX's/MOO's/etc. A play on symbols for the Unix generic string symbol, "*," which can mean any number of letters.