There could be a few reasons for this. Do you actually hear the system fan start? Do you know power is getting into the computer, led lights come on for the second it is on, you hear the system fan start, the you can hear the hard drive going? Do you see anything load on the screen? For example, if it is a Dell, you would see the Dell Logo come up for a second. I had this happen once on my computer and what was happening was my proccessor was fried, thats when I learned to wear tennis shoes when working on my computer ;) and make sure I was grounded so no static electricy would fry it.
My first thought would be power supply. Sounds like a bad lead on the main line that connects the PS to the motherboard. It could be the board itself or the processor, but that's more likely to shut down the computer or lock it up at the post screen than continually reset like that. If you have access to one, hook up a power supply tester. Use one with an LCD display that can tell you the voltage of each line. The basic ones will tell you "it works", but it can't tell you if, say, one of the 5v lines is bad.
If you don't have a tester, it's just as easy to just replace the PS. That's going to be cheaper and easier than replacing the board and/or processor. For me, if I'm replacing one, I usually replace both. Just make sure your old memory fits or that has to be replaced too. Hey, if nothing else, it's a good excuse to upgrade!
Just to be on the safe side, before you disconnect anything, take a look at the board. Look at the capacitors. They look like little tin cans wrapped in (usually) blue plastic with the tops exposed. You'll usually see at least a few gathered near the processor. The tops should be flat, generally marked with an indented "x" shape. If they are bulging or ruptured (solid brown crust coming out), then your board has about had it. Even if it runs, it's short on time. Replace it.