I was playing a game (WoW Beta), and suddenly after about an hour of
playing, the computer randomly shut itself off.

Pentium 4 1.4GHz
256megs ram
GeForce 4 Ti4400
Windows XP Pro SP1

I Can't turn the computer back on. It has a switch on the back of the
power supply to supply the electricity on/off and a button on the
front to turn on/off the computer. The button on the front wont turn
the computer on like it should, but I dont think the power supply is to blame,
because when I turn the electricity from off to on, the case fans turn
a bit, and the green light I have on my motherboard is on. I dont
think overheating would be a cause. I have run more resource intensive
programs on it before than a game.

I Dont have the time to spend working on it like I used to, so I am
pretty desperate for an answer I guess.
Thank you in advanced.

turn of the switch on the back , unplug the power cord , count to 15
slowly , plug in the power cord , turn on the power switch , hit the
front switch , if there is no damage to your machine it should come
back on again

I tried that multiple times, and  nothing more than the green light on
the inside comes on. There are no burned smells or melted parts that
are at all visible to the senses. I blew out all the dust with an air
compressor, and ensured all the cards are seated properly, and all
cables are connected correctly also. I'm stumped =/

try resetting the Cmos

I have an Intel D850GB motherboard - both CMOS chips are soldered =/

Does it beep? Does the fan turn?

There are no beeps, When I put the switch on the power supply from off
to on, the green light on the motherboard comes on and stays light,
and the fans turn for about 1 second, then is dead. Pressing the
button on the front to put the computer on does nothing.

I just tried testing for shorts by unplugging all the components
except for processor and memmory.

I reseated the memmory and processor. 
I looked for blown capacitators, and there are none visible... hummm

if fan only comes on for a second, the powersupply has failed and your
machine needs service.  what brand PC ? Is it under warranty?

It is a home made PC I built a year or two ago. I Leave it on all the
time because it is an FTP server. I suppose the problem is in the
case/power supply/motherboard. So I bought a new barebones system that
will not only probably fix this problem, but upgrade x3. 3.0GHz, 1gig
memmory, butt ugly case, but that should work. I'll just add the
accessories to it.
I think what happened is that something on the motherboard died, not
the whole thing, just a small, necessary, part.
If this new system doesnt work either, i will have to look more into a
possible shorted card, drive, etc.