A MUD (Multi-user dungeon) is usually a hack and slash style game, where you create a character with statistics under a certain rules system (official or original) and go around killing monsters and stuff to gain experience and level, items and prestige. Though there are different styles, and many of them are based in certain worlds or realms, from specific things like Forgotten Realms and Star Wars to simply generic realms like 'fantasy' and horror', as varied as most of the games here.
The difference is that they're fully text games - each room and each monster and stuff like that are all described, think a roguelike game like ADOM or Angband. And each room is connected to the others, you connect to the server using a console and wander around with (for example) north/south/east/west commands.
Some are non-roleplay at all, some are statless and fully rolyplay-enforced, it depends on what you're looking for. MUSHes are the other way around, they're also text rooms, but the underlying code structure is different, they allow for longer buffers and things like that in order to make it more condusive for roleplaying, while totally taking out any *automated* hack-and-slash monster killing element.
You connect to them using a console called a mud client - telnet is the simplest one that most Windows machines have, but there are a lot more elaborate, customizable ones.
Hm. Not explaining this very well, am I? Try the Wiki on it: http://en.wikipedia.org/wiki/Multi-User_Dungeon
and the links off of there. :)
Maybe examples are easier. I've played in a medieval fantasy hack and slash type game called Medievia - one of the largest around and with really, really good code behind it. There movement is based off of north, south, east, west, up, down, and there are 'zones' that people build (and the gods approve) for you to explore either solo, or as a group, with intense cooperation. No roleplay there at all (there is a background story that shuffles along like a turtle) - but you have overhead maps and such like an old-style Ultima game when outdoors, dragons that fly over the land and ships that sail across the water, maps that rotate depending on what you're doing, all sorts of special mob programs (Artificial Intelligence) to make them less predictable and harder, and tons of rare items. That's an extreme example of a MUD. You have characters with stats, four classes to work through (mage, cleric, warrior, thief) before you hero, and you need to eat/drink food and drink now and then. Hundreds of players on at a time though, and in some zones they player-kill heavily.
Then I've played on another place, a MUSH (Multi-User Shared Hallucination? Not sure.) called Winter's Edge, based fully around an original fantasy land but using D&D 3e (3.5e? I dunno, I left before 3.5e came out) rules. You create a character, but in the sense of filling out a character sheet, with nifty commands that save your information to memory but don't otherwise do anything.. it's not automated. You create elaborate background, description, and things like that, then apply for a character and a roleplay god will sit down with you and hash out details of your character, help correct any mistakes, then allow you onto the 'grid' (game). There are no automated monsters and all you can kill when bored, but you land into a town, walk around, and interact with several dozen other people online in poses like what we do on forum posts here. And quests and events are run sometimes, some spontaneous, some which are planned far in advance. The gods/storytellers run those, think GMs to be like the moderators in each of the games here, but they work with each other to attempt to create a free-flowing game.
Some are entirely statless, like a place I used to play on called Threshold MUSH, it was a horror/bdsm style game, you had to make description(s) and a partial background and personality, but you start in a large haunted house with everyone else that you all can never leave, some aggressive, some victims, and so on, and the creepiest things would happen when the gods were having a good evening and creeping random players with scary stuff like blood and flying ropes. :)
Except both are done in real time, which is the main difference between forum-style rp and MU* style rp.. and especially in MUDs, lag at the wrong time or a split second difference can have dire consequences if you're doing something dangerous. MUSHes are more freeflowing, the gods just use your character sheets to judge what you can and can't do and roll dice when needed and so forth. But both above are just examples, and rather extreme ones, it's not a black and white thing and even many MUDs mix hearty roleplay with automated text battles.
Hmm.. I need to put in a plug for Elliquiy though, I think I overdid that a little. The advantages of here.. it's well moderated, less immature people (some MUDs are bad for that), you have the time to think things over and take the time to pose so the average level of posing is MUCH better, you can quit a scene if you don't feel like playing it anymore, and move on.. the longest pose I've ever seen on a MU* (generic term for the whole group) is like fifteen to twenty lines or so, and by far a rarity.. most are 1-6 line blocks. Plus roleplay can sometimes get boring if you're in a large crowd and it takes forever for people to pose so it slows everyone else.
Wait, that happens here too.
Oh, and you can't play more than one character at a time, though most let you own multiple characters. But since you have to be online on one to get to do any sort of stuff, there's far less flexibility than forums.
Most adult ones are MUSHes, for the simple reason that killing monsters isn't very sexy. There are some that mix both. But a lot of the adult ones have very little moderation in the sense that it's very easy for immature people to slip on. Ugh at Shangrila for that.