if its not switching on at ALL, check its plugged in
check the connection between the power lead and the motherboard inside your computer (just make sure its firmly connected) and try using a different kettle (power) lead between the wall and your PC. most monitors and printers etc all use the same power cable.
It could also be the wiring from your power on button to the motherboard - there should be a small button on your motherboard which you can press to switch power on - should be right next to where the power supply connects to the motherboard.
If that doesnt work, you will probably need a new power supply. £20 ish. bish bosh.
If the worst comes to the worst (which it probably wont - power supplies fail all the time) and you want to save your data without people seeing it, you can buy a lead that converts an internal hard drive to a USB mounted external drive - the kit is called a "USB to IDE 3.5inch" or "USB to SATA 3.5inch" depending on the what writing is on your hard disk, that kit is about £10.