DMX is a digital control protocol written to control lighting fixtures.
Originally, a dimmer pack and control desk were the same piece of equipment, and a new cable had to be run from the control desk all the way to wherever the light was positioned. This was costly, not only in copper, but also in power loss over the distances involved (think large theatre).
So someone invented a digital control system, called “digital multiplex” which provided 512 different “channels”, where each channel could have a value from 0 to 255. This allowed the dimmer packs to be smaller, and placed much closer to where they were needed.
Nowadays, we have intelligent fixtures which are capable of doing cool things, more than dimmer packs simple dimmer controls. Each fixture listens on a start address, and for a fixed number of addresses after it. The number of addresses a fixture needs can normally be found in the user manual for that fixture, along with the mapping of what channel does what. For example, a moving head fixture may have channels for brightness, colour wheel position, gobo wheel position, pan, and tilt. Address configuration on channels is normally done via a DIP-switch or a control screen (depending on how new or advanced the fixture is).
DMX signalling is done as a daisy-chain from a controller to each fixture in turn. Controllers can be anything from control desks like the zero88 Leapfrog, USB DMX boxes like the ENTTEC boxes, or even a fixture in “master” mode.
DMX is transmitted over 5-pin XLR connectors (pins 4 and 5 are unused), but a lot of manufacturers (especially on lower-end products) drop to 3-pin XLR connectors. The majority of our equipment is on 3-pin XLR. The only exceptions are the lighting desk and dimmer packs.
For more information, consult Wikipedia