Cause 3: The power supply is broken

This chapter only applies to desktop PC’s, for portables there is no standard power supply so it depends on your model, however the general troubleshooting principles apply.

To Test basic functionality. Disconnect the power supply from the motherboard. Take a paper clip and short pins from the green and any black terminal of an ATX power connector (largest connector, typically 20-24 pins total) to power it on. Don't leave the paper clip connected, just short it until it powers on. Check to see if the fans spin (ie: inside the power supply). If so, your power supply is providing a 12V supply. An obvious clue of component failure is faint smoke or a burnt smell. Source: http://www.wikihow.com/Diagnose-and-Replace-a-Failed-PC-Power-Supply

There are also power supply tester available to be absolutely sure whether it is defunct or not.

If the power supply is defunct, replacing it is not that difficult. First disconnect all external wiring and open the PC case cover. Next you have to disconnect it from your PC, the power supply looks like this:

When disconnecting write down all the connectors you disconnect so you know what connectors to reconnect with the new power supply. Unscrew the four mounting screws on the back of the housing. Go to a PC shop with your old power supply and ask them if they have a new compatible power supply, usually this is no problem as there are not many standards. Install the new one in your PC and reconnect all connections you have written down previously. That’s it, no software configuration has to be done to be done.

For a detailed procedure on how to diagnose and replace a failed pc power supply, check out this Wikihow article:
http://www.wikihow.com/Diagnose-and-Replace-a-Failed-PC-Power-Supply

If your PC has a custom case, you can't just replace it with a generic power supply. Here's how you can use a multimeter to test key power supply components to repair a power supply yourself: http://articles.techrepublic.com.com/5100-10878_11-5034541.html