If you really want to know which component is dying on you should stress it individually... start with the ram, then the CPU and finally the hard drive(although be careful of destructive write dests).
1. Identify if it is software or hardware first, try running a Live CD like ubuntu, is the desktop stable after 10 mins use?
2. Lets assume hardware..Boot from a diagnostic CD/USB/floppy and run tests on each component.
3. If the above passes boot up a stable build of windows and run some real torture tests(see below), you could have drivers fighting over something and all you need to do is
Eurosoft's Pc-Check is a good tool to look at as it will run through a whole lota checks for you from CD or floppy(this is to rule out the OS or build firstly), my personal experience of it in terms of processor stressing though is it takes a while longer to do its job. stressprime has overheated CPU's for me in 10-15mins where as PCCheck took bout 30min(version 6.0 tht is) short of that you'll need to use something like “Hirens Ultimate Boot CD”..., here is a good time to run memtest as well as any other apps you can boot to in testing. e.g. I had a faulty RAM module and though I thought the build was fine after 1 month whenever I sat down to do my expenses the cumulative errors had built up to a point where the PC was just unstable(no kidding it was like clockwork and took me 8 hours to get my build the way I like it...each month till i worked out what was wrong and for some odd reason memtest didn’t pick up the error when all sticks were in but only when tested individually..at least thats what i remember)...like wise I have seen this happen for HDD's where they aren’t able to read a sector but windows thinks it can do both.
Ok .If you are able to get into a windows environment on stock settings install "speedfan" (the latest versions have a SMART short/long offline test as well as comparing stats for how your drive should be running)so you can see your nominal (get CPU-Z and look up what nominal voltages/temp ranges are for your CPU)temps and voltages, then start hammering the CPU cores with stress prime(and monitor, i have seen individual cores fail this). If stress prime goes through and everything is stable for bout an hour or so, consider the other options, depending on the board/cpu try a combination of stressing the CPU and the RAM, this will get the Mb's memory controller into the mix and heat that up too, again watch for stability issues...then there is what sometimes acts as the final nail in the coffin, graphics card..gpu-z and fur mark...if that all doesnt make it fall over at a particular hurdle I really cant think of what might at this point. Hell you could try and run all that at once an see what happens, CPU’s normally heat up very quickly an major fault come up quite fast. Subtle problems with them take a while longer...if its the graphics card you’ll get graphics corruption pretty quickly again during the test indicating either overheating/overclocked/faulty VGA card, once you stop these tests windows should be able to revert back to its normal snappy self. If it doesn’t ...Testing by inference you can normally pin it down..
Lucuis is right though..first step, write down the error code, as useless as it can be sometimes and see if it relates to anything that may help diagnose the problem....also one last thing with CPU’s, if the thermal cut off in engaged due to over heating (ahh when the water cooling pump died after 3 mins of use on an overclocked athalon 2600) you may have to wait a while before the cpu/mobo decides its safe to turn back on if you reboot as you say you did at least that seems to point to it not being a thermal issue..
just take it one at a time.