MUD, or Multi-User Dungeon/Dimension/Domain, is a most basically a term for a text-based multi-user game. Of course this is a gross oversimplification, but one which captures the essence of what is to follow.
MUDs have mostly been replaced by Massive Multi-Player Online Role Playing Games (MMORPGs) like Everquest or Lord of the Rings Online or World of Warcraft. MMORPGs are actually direct descendents of MUDs, although evolution in the computer world is much more dramatic than in nature. MMORPGs have a graphical interface while MUDs are generally text-only (there are some hybrid graphical MUDs, but they don't figure into this explanation).
Most people who have heard of MUDs tend to think of a sort of hack and slash text adventure game. If the MUD base in question is Diku or LP, they are correct. MUD is, however, an umbrella term which covers much more. Andy and I met on a type of MUD called MUSH, for Multi-User Shared Hallucination (some say Multi-User SHell). MUSHes tend to be more for roleplay (RP) and social purposes. In our case, the MUSH was primarily for RP, with some social interaction built in.
Many, but not all, MUSHes have themes. Themes can be a set of published books which MUSH players have all read, or based on some setting imagined by the MUSH founder(s), or a pre- or post- story which ties into a movie or song. They can be a re-creation of some historical period, like ancient Rome or Medieval Europe, or an exercise in historical fiction, along the lines of "What would have happened if one side won a war which they lost in real life?" They can be realistic, fantastic, based on Earth or other planets/galaxies, or in other dimensions or planes of existence. Basically, if it can be imagined, a MUSH can be themed after it.
The cool thing about MUDs is that you're interacting in real time with real people from anywhere in the world. When you type something in and hit enter, everyone in the same virtual room that you're in sees it and can respond. RP MUSHes are much like interactive playwriting or storytelling in that you say what your character does, and others respond in the way their character would. In that way in fall 1992 a senior at the State University of New York (SUNY) college at Oswego ran into a high school senior in Indiana and struck up a friendship behind-the-scenes on a themed MUSH.
Tomorrow I'll be describing Pern, which is the theme of the MUSH on which Andy and I met. It is a fictional world by Anne McCaffrey, written about in some dozen books which Andy and I had both read before we started playing on the MUSH.