I did some exploring on these terms and this is what I discovered:

MUD refers to computer gaming.  It usually involves fantasy role playing.  For example, if you visit a MUD, the fantasy world may be made up of elves, dwarves, or goblins.  As a player visiting a MUD, you may be able to take on the role of a warrior, thieve, or priest.  As the game progresses your goal would be to acquire more skills or powers for the role you are playing in the game and to slay monsters, complete a quest, go on an adventure, etc.  Players interact with other players during the game.  The game is text driven, meaning that if you want to know what a room looks like, you have to read a description of it; there is no picture. 

A MOO is similar to a MUD.  In fact, I couldn’t find much difference in them.  A very popular MOO is LambdaMoo. Basically, LambadaMoo is setup like a house.  When you first enter the house, you can choose to access the Coat Closet or the Linen Closet.  From there you can enter the Living Room where many people hangout and talk to each other.  While in the Living Room, you can roast things in the fireplace. If you sit on the couch, some of your objects may get lost and fall underneath of the couch.  There is also a pet Cockatoo that repeats phrases he hears the players say.  It’s not hard to see why at times people use cloth to bandage the Cockatoo’s mouth!  During holidays, the Cockatoo is exchanged for other birds (a turkey at Thanksgiving, a Raven at Halloween, etc.)  There is also a hottub, a pooldeck, open fields, fishing holes, library of online resources, etc.

MUSH also sounds very similar to MUD and MOO……..it basically refers to computer gaming that can be conducted online; it also uses role playing and allows players to communicate between each other. One difference I discovered though was that players can create their own rooms, so it allows for a little more creativity.  It seems like of the three, MUSH has the most capabilities.