Since a lot of folks that never used one have no idea what a MUCK is at all, I'll try to sum it up.
A MUCK is basically a text-based world, in which players interact in form of some character they create. The character can then walk around in the world, look at stuff, examine things, and most of all talk to other characters that are in the same part of the world he/she/it is in. Depending on the kind and flavour of a MUCK, there are certain themes that are kept up and commands that assist interaction or navigation, that vary widely from place to place.
(For an introduction to Faibanx MUCK check out the hompage or go here: http://lavadomefive.com/cs/forums/p/131/829.aspx#829 )
Despite the commands that assist you, interaction in a MUCK is all about describing. You can think of it as a multi-player text adventure with lots of roleplay, a net of connected chatrooms, a virtual city, a community home with mail function, and a lot in between. There have been some attempts to put graphics into MUCKs, like Furcadia, and I would also count secondlife as such an attempt, but having to put everything and every fantastic character concept into a metric space of polygons and sprites and put all interaction into animations greatly reduces the possibilities a text MUCK provides.
Furry MUCKs, like Faibanx MUCK, focus greatly on the furry characters and the interaction between them, which is often of a romantic or sexual nature. When being 'In-Character' (IC), people expect you to be, act like, and behave like your character, and all of those MUCKs have (as far as I know) a system to state your character's sexual preferences, although some players prefer that to be found out during the interaction or asking Out-Of-Character (OOC). The amount and degree of mature interaction, just like the strictness of separating IC or OOC topics, varies from place to place too. While Faibanx MUCK for example is clearly for adults only it focuses on free mature and open-minded interaction and conversation. That CAN go whereever the players decide to take it and end up in the roughest and most perverted macrophile orgies, but it doesnt have to. There are other MUCKs where people only connect to to have online sex, and yet others where mature talk or certain kinks are forbidden in public areas.
Hope that helps, please correct me when you think something wrong or could be said better.
Wikipedia says: http://en.wikipedia.org/wiki/MUCK, WikiFur says: http://furry.wikia.com/wiki/MUCK