Short for Multi User Dungeon, it's a world you can explore and interact with alongside other users. They were generally text-based (like Zork), though I believe there were some roguelikes and graphical MUDs too.

Many multiplayer online games played today have a heritage that comes from the MUDs. Minecraft is certainly one of them and fills a similar niche, though to categorize it as a MUD seems strange, unless any multiplayer virtual world with an ROG-like element is also a MUD.

The link in the grandparent comment is (essentially) a really good and extensive definition of the term. And then also a history of MUDs and what they've since evolved into.