What Does it Do?
DMX (Digital Multiplex) is a Control Protocol (think of it as a language) that you use to communicate with automated fixtures from a control console (like a Hog or MA). This protocol is not manufacturer specific: DMX is globally embraced by manufacturers of lighting fixtures and consoles. DMX can carry 512 unique channels (sometimes called "Slots") of control.

The Cable
The cable is a twisted pair style (that's the internal wiring) with shielding. This cable has to be capable of 100 ohm or 120 ohm. The connector is a 5-pin XLR. This does not mean microphone cable! Microphone cable is not that same as DMX cable. Although you will probably run into people or installations that use microphone cable- this does not make it correct. Microphone cable used as DMX will fail; and given Murphy's Law: it's gonna fail at the worst possible time. Don't do it!  Don't believe me? USITT has the technical reason why you shouldn't do it:

While there is some tolerance allowing for 100 ohm to 120 ohm cable, simply put, microphone cable is not at all suitable because of its high capacitance and incorrect characteristic impedance. It might work in some instances, but it is not appropriate for the electronics involved and it will fail at the most inopportune times.

Hey, that last part sounds familiar, huh?

Connecting the DMX Cable from Console to Fixture:
The male end of the cable gets connected to the control console's output port. The female end will be connected to the "Data In" port on the first fixture in line. From the fixture, connect a second DMX cable's male XLR connector to the fixture's "Data Out" port and run to the next fixture in line. Follow this idea to "daisy-chain" to the rest of the fixtures on the universe.

My fixture has 3-pin XLR connectors: Why?
It's likely that the fixture was manufactured before the DMX512 standard was made official. Most likely, you can still control it. You can use a 5-pin to 3-pin adaptor, but keep in mind that each time you adapt, you degrade the control signal. So, try not to place your fixtures in a manner where you are daisy chaing a 5-pin fixture to a 3-pin fixture to a 5-pin fixture to a 3-pin fixture....
Oh, then there are the 4-pin fixtures. Lots of scrollers use this style. Most often times, the scrollers will have a PSU base station that will accept your 5-pin DMX cable. From the PSU, you will connect the 4-pin data cables to each of your scrollers.

Termination
The last fixture in your DMX line should always be terminated. This can be achieved with a simple 5-pin male XLR connector wired to match the impedence of the DMX cable. Internally, pins 2 and 3 are soldered to a 110 ohm .5 W resistor (see a diagram of the wiring from Blue Room's Wiki). While many venues neglect this last piece of the chain, it is a requirement of DMX 512 protocol, and not using one could result in strange activity on the DMX line. Just like using microphone cable, not terminating will result in system anomolies that will present themselves at the worst possible time.

Universe vs Channels
A DMX Universe is a group of 512 discrete channels. A channel is one parameter of control: one channel could control intensity, or a gobo selection, or a CYAN flag... well, you get the idea. Each channel is capable of 256 discrete values (0-255). For example, to call up a particular Gobo, a hypothetical fixture may need to see a DMX value of 152 on channel 13 on universe 1. On the console, this is usually done as simply as touching a button corresponding to the name of the Gobo you wish to call up.  

How does the address of the fixture work with DMX?
Every fixture in the DMX chain will need to have it's own unique starting address. Let's consider a hypothetical automated fixture that we'll call "Supa Dupa Spot."  Let's say that "Supa Dupa Spot" is a 10 channel fixture. That means that the fixture has 10 unique attributes you can control (Examples: intensity, pan, tilt, gobo, color, strobe, etc.). You wouldn't want to affect intensity at the same time you were trying to pan the fixture, the channels must be unique. On the base of "Supa Dupa Spot" is an LCD menu where you can enter a starting address. This address will be the DMX channel for the first channel of control of the fixture. If you were to address this fixture at channel 1, then channels 1- 10 will be understood to control "Supa Dupa Spot." If you are connected a second "Supa Dupa Spot," then the first available address for this fixture will be channel 11 (as channels 1-10 are bring used by the first fixture).  

After you've addressed the fixtures, you will need to patch the units into your automated lighting console. Here, you will need to tell the console the type of light that it is ("Supa Dupa Spot") and also which universe you have connected the fixture to, along with the starting address that you've entered on the fixture's menu system.  Modern automated consoles contain "Fixture Libraries" which allow the console to know exactly how many channels the fixture uses and the order of each channel (and all values for each channel) for the fixture. While I realize this may sound complicated, the good news is that you don't have to worry about the order of the channels or the discrete values for all of the parameters: the console will have that information.  

Learn More:
USITT, the developers of the DMX512 standard, have a great FAQ page available that will get you even more in depth with DMX.
Also, check out Doug Fleenor's DMX Primer.