MUDs (historically Multi-User Dungeon, with later variants including Multi-User Dimension and Multi-User Domain) are a broad category of multi-user virtual world systems -- usually text-based, usually games, usually accessed via the Telnet protocol, and usually free. They date to Roy Trubshaw and Richard Bartle's creation of MUD1 in 1978, and are the direct predecessor genre to the MMORPG.
The MUD genre peaked in popularity in the 1990s with the spread of the four major MUD codebase families: AberMUD, TinyMUD, LPMud, and DikuMUD. All of these codebases, though open-source, were created with non-free, anti-commercial licensing, which guaranteed the availability of free games in the short term but hobbled the evolution of the genre in the long term. MUDs had the greatest relevance when they were essentially the only game in town if you wanted multiplayer persistent-world online RPGs; once MMORPGs (originally referred to as "graphical MUDs") came on the scene in the late 1990s, MUDs were quickly eclipsed culturally and suffered considerable shrinkage of their market, which has never recovered to its mid-1990s levels.
Despite these setbacks, a number of significant MUDs continue to operate, even including Trubshaw and Bartle's original MUD1 and the first LPMud, Genesis. New MUDs continue to open as well, though it is notoriously harder for them to attract attention and a workable player base than in earlier days. The genre is also worth noting for the tremendous amount of innovation and exploration of the virtual world problem space that it has produced; it's not for nothing that Bartle's Designing Virtual Worlds, a volume based largely on experience with MUDs, is called "the bible of MMORPG development", and many luminaries of MMORPG creation, like Raph Koster and Brad McQuaid, began as MUD developers.