Warning! There are several large capacitors in ATX power supplies, that will store a dangerous charge for a long time. Please let your power supply discharge, completely unplugged from the wall outlet, for a few days before opening it up. You can probably be seriously hurt, so please be very careful. I am not a lawyer, but I hereby release myself from as much liability as I can, for any sort of injury you sustain, or any trouble you get into.
Step 1: Background
Computer power supplies are Switch Mode Power Supplies (SMPS), which use high-frequency switching circuit elements to provide a high-quality output voltage, with good energy efficiency. One side effect of this technology is the minimum load requirement that each power supply has. In order to function properly, the power supply needs at least a very small electrical load connected to it. In other words, ATX power supplies will only work if you have something connected to it. We will be using a power resistor to provide this minimum load.
Also, modern power supplies do not simply have an OFF/ON switch, they have what is known as a "soft" power switch. This normally makes no difference to the user, as the computer behaves the same, but when you shutdown your computer, the motherboard can turn off the power supply when it has finished shutting down. This requires us to add our own power switch to the power supply chassis.
To protect our circuit from accidental (and careless!) short circuits, we will install some fuse-holders and fuses, which will disconnect the circuit supply lines if too much current flows. The size of the fuses are up to you, but a 1 amp fuse will work just fine for most circuits. You really should put fuses on all supply lines.
Update: While the diagrams show fuses on all voltage rails and no fuse on the ground line, when I actually built my power supply, I was young and foolish and only put a fuse on the ground wire. It's much safer and a better idea to put fuses on all signal lines and not the ground line. Thanks to many emails and messages on Instructables about this oversight.