Make sure all of these are turned on:
1. Make sure the power supply voltage switch is set correctly. The position the switch should be in depends on the country you’re in. If it is not set right it may not power on at all, or could cause permanent damage. Check this chart to find out the correct voltage for your country.
2. Check for a loose or disconnected power cable. You would be surprised how often that will fix it.
3. Replace the computer's power cable with an identical one from another computer. This is just a test. If it is bad you can purchase a new one easy.
4. Check the wall outlet. Plug a lamp or something similar into it.
5. Test your power supply.
6. Replace your power supply
7. Replace the power button on the front of your computer's case. It’s a cheap part, and saves you the trouble of testing it. You want to test it?
Trace the wires from the power-switch button to the pins on the motherboard. Then pull the leads off the motherboard (remember where they go) and check the switch wires with a torch-bulb and battery, continuity-meter, or multimeter. If that doesn’t work try bridging the motherboard connection pins with a short length of wire for a second or so. DO NOT touch anything else around it with the wires! Damage will occur.
8. If none of this worked it’s likely the motherboard will need to be replaced