My computer has been doing this for a while now. Every day at least once a day it will turn itself off. The power button and keyboard remain lit but the machine stops functioning. When I then press the power button it beeps. I have to turn it off at the wall and wait a couple of hours before turning it back on or it won't start.

It seems to happen more often when I run high graphic games but most of the time it happens randomly.

I called someone in to take a look at it and they said it might be the motherboard or power supply but I was wondering if there was a way to tell for sure. It's not other computer parts because I've tried unplugging each of them and seeing if that helps. It also isn't the monitor because I changed the power cord and the machine doesn't turn on even without the monitor.

Something is overheating, it could be any number of things including the CPU, the system wouldn't beep if it was the power supply. Thus the reason it starts up when cool, in fact I'm quite sure it's the CPU or Central Processing Unit. Depending on the age of the computer, the heat distributing paste between the CPU and fan-heatsink degrades and stops carrying heat away from the CPU. When the CPU reaches a certain temperature threshold, the computer shuts down. Let me know if you require any further assistance.