MUD is the correct term. It's short for M(ulti) U(ser) D(ungeon). They're mostly for roleplaying.

These are different from MUCKs, which are short for M(ulti) U(ser) C(arnal) K(nowledge): despite the title, many are not. Some have roleplaying aspects but it's not required.

A third term is MUSH, which is M(ulti) U(ser) S(imulated) H(allucination). I think they are simular to MUDs.

Very confusing.