Welcome to my MUD list.
A. A MUD is a multi user dungeon. Basically think of them as MMOs before MMOs existed. They were (generally) entirely text based. You'll sometimes see them called MUs or MOOs as well, but mud is the most familiar term. There are tons still running too, and new ones even get spun up.
A. Because they're fun. Since MMOs have existed for about 25 years now, MUDs had to distinguish themselves somehow, and most of them did it through either a focus on enforced roleplay and/or on user-created content through powerful scripting systems. They're a very different feel overall, especially in a post-WoW world where pretty much all the old trappings of MUDs were shed.
A. Some of MUDs offer web clients. I recommend that you get yourself a copy of either tintin++ or wintin though. I've also used gmud in the past and I think it had good results? Then you need to find a good MUD for you. There might be some on this page. Failing that, consult your search engine. mudlistings and Mud Connector are good places to start.