I know this may not seem very technical, and no offence, but it doesn't sound like you have an A+ certified tech around so... you should find another computer, and temporarily replace each part of the old computer until it works.
I.E. put in working computer's Power Supply Unit in old computer. If that doesn't help, put in working computer's RAM in old computer, etc. You will eventually find out which part is faulty through process of elimination, and in this case it might be the motherboard as some people stated. However, do NOT be hasty to buy a new computer immediately, unless the upgrade is really justified (I justify my upgrades every month by saying I really "need" that nuclear powered video card!). 99.9% of the time, the issue can be resolved by replacing one part, or something as stupid as taking out and then plugging in all the internal components again! Trust me, I have literally months of experience tinkering, and more than half the time, something random occurs that brings things back to normal!
edit: Do exactly what scooter said above in fewer words...