MUD stands for "Multi-User Dimension" (D&D fans sometimes refer to it as "Multi-User Dungeon" because of some of the similarities between MUDs and D&D).
A typical MUD allows as many as 20 (sometimes more) people from all over the Internet to be simultaneously connected to and playing the game. When playing a MUD, you are able to interact real-time with other players. Because of this, new players often view MUDs like being connected to a "chat room" on a BBS.
Here is where the BBS similarity ends, however. A MUD has the added dimension of simulated physical space. You have a body with D&D-like statistics (strength, dexterity, wisdom, and the like). You can explore the MUD world by issuing movement commands (like "go north", "enter pub", or just plain "w" for "go west"). In this respect, every MUD is unique because every MUD has its very own geography.
Additionally, in MUDs you can encounter hostile monsters that you can engage in combat. If you win the fight, your character will gain experience points that can be later spent on improving your body's skills and statistics. Or, you can allow your experience points to accumulate until you are eligible to advance your character's level. When you first start the game, your level is 1. Most MUDs allow you to advance to level 20, at which point you become immortal and are eligible to become a MUD programmer.
MUD programmers are called "wizards" on most MUDs, although they're called "Coders" on Core 2651. A wizard/Coder writes simple programs that can be used as rooms, weapons, armour, or monsters. These items then become permanent fixtures in the game. In this way, a MUD programmer has the ability to literally change the geography of the MUD. This is why no two MUDs are ever identical.
If you have never played a MUD before, but are a bit timid about diving right in and getting lost, don't worry. On most MUDs you can quickly find another more-experienced player who is willing to show you around and get you started. (Most of these "newbie-helpers" might even give you some starting equipment and money you can use in the MUD!)
Core is a game of the 'MUD' genre.
You can reach Core by typing
telnet coremud.org 4000