Make:

There are a few voltages in PC power supplies. The older ones I've worked on most, had 5V and 12V outputs. Newer ones have lower voltages also. On the older ones, the 5V had to supply around 20Amps. It takes several 18AWG wires to carry that current without voltage drops causing trouble. Typically, the highest power output is the one that is used to regulate the voltage, and needs to be loaded with a resistor if the load you want to power isn't big enough. I'd start with 20% of rated output. If you are tapping into a whole computer, just use one of the disk drive cables, or get a splitter that would allow for a second drive.

Hacking it like a second disk drive is brilliant! Would that carry 12 volts? It wouldn't need supply more than 3 to 5 amps, would I need to protect my electronics from the power supply?

The regular disk drive cables (for IDE drives) have 4 pins. The 2 center (black) wires are ground. Best to double check with a voltmeter, but I think the yellow is the +12V.

The current rating for each voltage should be on the supply. A couple of random PC supplies I have laying around rate the +12V at 8 amps max. Make sure the MB, and other disk drives are leaving enough with some room to spare for your add on circuitry. The 12V won't be tightly regulated, and I'd expect some noise on it, but add a small inductor rated for your load current and some caps to the side going to your circuitry, and it should be ok. At work I have been involved with the design of audio circuitry that we add into PCs using the extra disk power connector to get power.