A MUD is a multi-user dungeon. It is based on dungeons and dragons, a roleplaying game. It was one of the first ever uses of the internet.
A MUD is a text based adventure. You have various skills and abilities, weapons and armour, as well as food and items that you can carry. You move from room to room with commands, and then you kill "monsters", which gives you experience, money and equipment. Once you gain enough experience, you gain levels.
MUDs have always had their group of people who play them, however with the advent of MMORPGs, that group has all but disappeared. Nonetheless, they remain popular within a group, as they can be played while doing something else. Because they are text based, they do not take up very much processing power.
There are all kinds of MUDs going about. Virtually every topic, every theme, every cartoon or movie, has had a MUD based on it.