Every MU* has an interactive interface where several players can log on and interact with each other online, and they all have some sort of game involved with them. The game is often roleplaying in a Dungeons and Dragons format.
Each seperate MU* (I'm going to use MUDs and MUSHs as examples as I am most familiar with them) has its own quirks and rules. For example, a MUD is more walking around and encountering game simulated monsters and destroying them. They have distinct objects as you can level and gain more points for your player. You also die frequently and have to start over. MUSHs are more of an acting experiance, and the goals aren't always clearly defined. Many MUSHs are games where you can attack other players in different factions, yet others are more peaceful and you fight only computer generated things. You can have more control over whether your character lives or dies in a more passive MUSH.
Depending on your preference, you can find yourself in various situations. You could be a warrior fighting evil demons, you could be a Rebel in an X-Wing, or you could be a dragonrider of Pern. What you do, and who you are is completely up to you.
Another aspect of MU*ing is online coding. You can create and code objects (not generally in a MUD though) and interact with them. An example of this is a communicator, such as on Star Trek. You can create the communicator, and then program it so you can talk to other communicators and talk to your shipmates. You can even limit its range depending on how your MUSH is set up.
The best advice I can give about learning about MU*s is to log into one and see for yourself. Most MU*s allow guest access so you can log in without creating a player to look around. MU*s are a very good example of how sometimes it's just better to learn as you go along.
If you have any specific questions, please feel free to mail me and ask. I'll be happy to help you start out somewhere. Who knows we might run into each other on Pern or on Bajor.