MUDs are on-line roleplaying games, and I have been head Coder (Zaylon) for one of these muds (Cities of Glory), and coding on other muds as well.  Depending on who you ask, different people seem to disagree as to what MUD really stands for, so I won't even try to tell you what it means.  They explain it somewhat on the link to the CircleMud site below.  Basically all you need to know is that these games are text based, and despite what the immortals on each mud claim, they are very much similar.  They are played over telnet, and are written, usually, in C/C++.  Each MUD has its own features, though they have many very similar commands, and once you learn them on one MUD, they are usually the same on virtually every other MUD there is.  For instance, one almost all muds, there are many classes, but four are always common.  These are Mage, Warrior, Thief, and Cleric.  These are the basic classes (basically jobs) in just about any game, and therefore when you learn about them, you can play them just about anywhere.
    MUDs are usually built upon a theme, and more often then not, the theme is medieval (knights, fuedal systems, etc..).  They not only take on this though, but also often take a fantasy approach (dragons, magic...), which is consistent with the more mundane Dungeons and Dragons.  Though each MUD will differ in the extent to which they follow their themes, and the extent to which they follow any real-world game (such as Dungeons and Dragons, GURPs, BattleTech, etc...)
    These games are typically what is called Hack-and-Slash in the gaming world, in other words they are about killing monsters, gaining experience from doing that, and gaining powers or levels as your experience increases.  Some MUDs are/have been moving towards a more Role-Playing atmosphere, and most MUDs are actually moving in that direction.
    In a Role-Playing MUD, one actually tries to play the role his/her character is in.  In other words if you pick a thief, then you generally want to be very opportunistic, able to be hired to perform certain tasks, and basically be a rogue.  Though the point is in solving problems (quests), or interacting with other players, but is more than generally just going out and killing monsters...  The idea behind this that you can play the former type of game at home, without a network connection, and the latter actually requires (wow...here's a shock) to talk to other people, or in some way interact with them.
    MUDs are not the only thing out there though, there are things such as Ultima Online, and MUSHs.
    There is a derivative of common muds, known as MUSHs -- this one everyone is pretty much in agreement over -- (Multiple User Shared Hallucination).  These games are played using many different types of clients, and as the name suggests, they are graphical.

•  Mostly, I have played on CircleMUD type MUDs.
• A very cool, and very old MUD is Cities of Glory (website)
• Dibrova (website), Another interesting mud, though unpopular because it is hard, and not very established.
• An interesting MUSH, CrystalMUSH

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