A lot of issues people have had with their computers can be traced to faulty or failing computer memory (”RAM”). There are several memory testing applications out there. MemTest86 is currently my favorite choice overall.
MemTest86 can be used to test your memory at its default settings, or you can use it to test your memory after adjusting its timing or if you are overclocking.
If you want to stress test or “torture test” your system, Prime95 is a good program. It will push your RAM and CPU to the limits, and will let you know if your system can currently handle the speed at which it is running.
MemTest86 (Personal Favorite!)
http://www.memtest86.com/
Prime95:
http://www.mersenne.org/freesoft.htm
(Prime95 Wikipedia)
Microsoft’s Windows Memory Diagnostic:
http://oca.microsoft.com/en/windiag.asp
DocMemory PC RAM Diagnostic:
http://www.simmtester.com/
(Page ID: 319)

MemTest86 is so good, it’s included in the Ubuntu Linux CDs now as a boot option.