A door game is a game that is played through a BBS, often further specified as a MUD, adventure game, strategy game, etc. The software is actually run and managed on the BBS server itself, which was nice on the programmers, because they did not have to worry about network programming, sockets, synchronization, or any of that junk, just interfacing with the BBS software.
The reason it is called a "door" game is, I believe, because the game is not run by the BBS software itself, but by another program, which interfaced with the main BBS software. So, by running the game, you were kind of "leaving" the BBS, and when you stopped playing the game, you "re-entered" the BBS, hence a door. There may also be some historical reason to it being called this, and I would welcome any insight into it.
What was revolutionary at the time was that this was most people's first experience with playing electronic games long-distance. Later on would come games like DOOM which also used BBSs to play games, and many other games of that era also supported playing through dial-up, but few were door games.
These were pretty much the driving force behind most people's interest in BBSs in the first place, right up there with the actual bulletin-boards themselves, file sharing, and email. But really, it was all about the games.
The most common games of this type were the Seth Able's Legend of the Red Dragon, TradeWars 2002, and MajorMUD, amongst many others. The games were almost all text-based, since they were played over modems with limited drawing capabilities: if you were like everyone else, you had ANSI color over a VT100 terminal emulator on a 80x25 MS-DOS screen in COMint, running over a 14.4kbps modem. Some rare games used RIPterm and RIPscript to display amazing 2-D graphics (pretty standard 16 color), but they always seemed to lack the imagination of their character-driven friends.
Unfortunately, the downfall of door games came around the mid-to-late 1990s, with the widespread use of Internet. With ISPs such as America Online offering unlimited dial-up hours, users had little need of their local BBSs and their small collection of games, along with hefty per-hour fees, slow connections, and limited phone lines. Thus was born the era of playing games over the Internet (see Battle.net, for example).
There are still many BBSs operating today, although many have abandoned dial-up for telnet access through the Internet. People's Connection, MariNet, Forceten are a few of these that are still up and running, where you can conjure a fine game of wandering through the woods, and being beaten to death by an old man with an ugly stick.