Other Ports: Although instructions have first been given for ZDoom, there are a number of other ports that are available:
Boom:
Chocolate Doom:
Doom for Windows 95 Launcher: Included with Ultimate Doom for Windows 95
Doomshel: From about 1995. Obsolete and doesn't work with Windows XP.
Doom Source Ports SVN Builds: Archive including some ports that I didn't mention.
Doomsday and JDoom:
DosBox 0.74: Good for Wolfenstein 3D and other games too.
DRD Team SVN Builds: Good for AFADoomer's conversions.
DWANGO: Wikipedia says Dwango ceased operation after 1998.
EDGE:
Eternity Engine:
GZDoom:
JDoom:
KDX Launcher:
Legacy: Also here.
MBF: Marine's Best Friend
PrBoom:
PrBoom+:
ReMood:
Risen3D: Recommended by Laz Rojas for his mods. The Risen3D download page has patches. Risen3D also has special extra Ultimate Doom and Doom 2 levels that come with it to take advantage of effects.
SDL: Klooni v1.3 SDL (Public Beta) is an example of an SDL game. This means that it has certain extra features and won't work in DOS or DOSBox.
Skulltag: Multiplayer. Contains extra episodes to take advantage of coding and effects.
Vanilla Doom: The nickname for using The Doom or Doom 2 command line parameters.
Vavoom:
WinDoom:
ZDaemon: Multiplayer
ZDL: Launcher by BioHazard. 3 Different Versions are available.
ZDoom:
Other: Amiga Source Ports and Emulators:

I know I'm forgetting one. If you can find one that's not listed on DoomWorld's Classic Doom ports, Doom source ports SVN Builds Archive, Doom Wikia's source port page, Planet Doom's source ports, Wikipedia's Doom source ports, ZDoom's Wikia of source ports, or FPS Ports, then add to the wiki or contact them. Check out FPS Ports anyways for source ports including for Wolfenstein 3D, Rise of The Triad, and others.