Computer won't turn on?  

Can you check the Power Supply, probably that cause your computer not turn on after you pressing the power button.

have you tried flicking the switch in the back its a small black thing that switches side to side it says 140v and 220v try this it might work it worked for me

Try to check the fan of power supply if its working,if not then probably there's a problem with the power supply...

This is a common phenomenon when you computer gets overheated. Do you have proper cooling system? Check you fan to see whether loads of dirts are blocking the fan blades or not. Since you are saying that your port lights remain on, that means the motherboard is getting power. Very slight chance that PSU has problem but still you can check for all the connections to the motherboard from the unit. Did you check you RAM? Because this can also happen if your RAM is somehow burned.

Actually there's a lot of symptoms for the problem. Get some basic checks for yourself and see if it works or else take it to a pc doc.