Steve pretty much summed it up. To elaborate further, if you want to go the simplest route you will need:
A computer running software that will handle the timing. Vixen is simple and free.
Something to get the control signals from the computer to the box that controls the lights. For this site, we use the Lynx DMX dongle or the Pixelnet USB dongle (more on that later).
You need the box that controls the lights. This is known as a dimmer box. This site supports the Lynx Express. One Lynx Express can control (turn on, off, and any intensity inbetween) up to 16 channels.
Finally, you will need the wires to connect everything together. USB cable to attach the dongle to the computer. Standard Ethernet cable to connect the dongle to the Lynx Express.
What I really like about RJ's equipment (he's the wizard that designed all this stuff) is it is all modular. If you want to avoid Ethernet cables strung all over and go wireless, then you build the wireless modules and plug them in and you are good to go. Need more than 16 channels? Plug in another Express.
The latest and greatest is the Pixelnet USB dongle. It is the replacement for the DMX dongle. It can do the same thing, but the added capabilities are unbelievable. In addition to working like the DMX dongle, with an added daughterboard called the Conductor you can run your program off a memory card and not have to use a computer. It will output both the controls signals as well as the music. There is also a Trigger board that works with the Conductor that has up to 15 triggers. Say you have a pressure mat under the floor, when the ToTs step on the pad, the trigger board can tell the Conductor to play a sequence that emits a scream and flashes the lights. I use the equipment here for both my Halloween decorations as well as for Christmas. The Pixelnet USB dongle needs a little more hardware, but all this is like potato chips, you can't stop with just one.
The Beginners page on the Wiki will get you familiarized with the terminology and equipment. Armed with that, you'll be able to follow along when we start reeling off acronyms and terminology.
Welcome aboard!