Thank you for the response. A lot of useful tips in there, although I'm not sure what "connector for the power switch to the mobo front-panel header"?
On a computer case, you have your power button on the front. This is a pushbutton switch which has, typically, two wires coming out of it, coming to a small black rectangle with two holes on it.
Many of the other parts, such as power LED, reset switch, HDD LED, etc, have similar things, though some use three, and it varies from system to system which ones you get.
On your motherboard you will have something called a "front-panel header". This is a term for a group of pins, usually 2 wide by some-number long, which are where these little connectors (the black rectangles) get plugged into. When you push the power button, it makes a momentary connection (or short) between the two pins for the power switch, thus telling the motherboard you have pushed the button, and triggering the power on, power off, standby, or whatever behavior is appropriate given the current state of the computer.
Shorting what two pins?
Could you possibly point out where the CMOS reset jumper is?
It varies from system to system. If you're completely unsure, you may be able to find reference docs for your motherboard online - if you built the system yourself or the system was custom built, the motherboard may have a manual. If it was a big-box build (Dell, Lenovo, what have you), then you may need to search online to find one of those, or call the manufacturer.
Note: Some motherboards work differently and do not have such a jumper or reset.
I hope it wasn't a short, more than likely lotsa things would be damaged (probably the mobo)?
When I plugged the cable in to the back of the PSU, the power LED only turns on sometimes. Most of the time it doesn't.
Oops... How embarrassing. I wasn't aware that doing such a thing could cause problems. I hope it hasn't resulted in/cause any major problems in the future.
If it was a short, there may be some bad power going on.
Again, are you indicating you plug the power into the computer and the LED on the front panel comes on, or is there another LED on your motherboard for power (some motherboards have it as a diagnostic/troubleshooting feature). Does it stay on or come on very briefly? Does it flicker?
Finally, doing ANY work inside your computer which involves connecting or disconnecting cables or power lines is BEST done with the computer off, the power out, and you being properly grounded with the case.
If you're unsure of how to do so or any other steps from here, my best recommendation becomes a local technician, who may be able to power-test the PSU or the motherboard.
While I dislike telling people not to help themselves, in this instance you're starting to see possibilities of electrical shock if you play around too much (it's very unlikely, but still).
I, unfortunately, am not an electrical engineer, or I could give you better advice - this is more diagnostic of "What happened" rather than "how to fix" -- because chances are if you're getting no response from the power button, and the section about the motherboard front-panel header doesn't help, and we can't find the CMOS reset, it's kind of a guess and chances are we're looking at hardware failure.
If you post the model of your computer here (assuming it's a brand name/big-box), somebody might be able to reply with the instructions on those steps -- I wouldn't know though myself.
Sorry for the disjointed nature of my post, it's hard to do when you're doing this on available time on the local service desk ;)