Characters have to be approved? hmmm, i think having someone else decide if the character I choose to play is allowed is a little bit controlling.
Never played any I'm not even sure what they are exactly if I'm honest but I've always associated them with telnet and BBS boards rightly or wrongly. I did a brief search there for my own curiosity more than anything and found this http://mudstats.com/MainPage.aspx Appears to list quite a few still about.
telnet, yes for MUDs. bbs ummm not as far as I know... do you mean games like ummm oh I forget the name... DND type dicey roleplaying-ish games?
There used to be an app for Legend of the Green Dragon, which was a very famous MUD back from the old BBS days (usually Red Dragon back then, but same game). I think it's been pulled since I've last checked.