MUSH stands for Multi-User Shared Hallucination and is essentially just a catch-phrase for a multi-user text-based game, like MUD (Multi-User Domain or Dungeon) is. MUSHes tend to be more role-play based than hack-and-slash like many MUDs out there.
On the technical side of things TinyMUSH is simply a code platform for the multi-user text-based games. Other code platforms include TinyMUX, TinyMOO, MUCK and various forms of MUD.