i'd try some of the real basic stuff before diving into the deeper things that the other posters have outlined.
First try what wuliheron said, see if you can hear any beeps. Those are error codes, the computer telling you what has gone bad. If you hear any look it up on the internet and it will tell you. You would google something like "post beep code 3 short 1 long" for example.
Try to reboot after each step to see if the issue is resolved.
- Unplug the IDE/SATA cables connecting your CD drive and hard drive from the motherboard (aka mobo)
- Remove the 4/8 pin CPU power connector (should be right next to your CPU) and reseat it.
- Remove the 20/24 pin mobo power connector (should be on the side away from the back panel) and reseat it.
- Reseat your RAM, if you have 2+ sticks, remove them and then just plug one in in DIMM0
- If you have a video card remove and reseat that as well (if you have onboard video, then see if you can remove the video card completely and use onboard video instead)
if after all of that your computer is still not booting, then most likely, the problem is one of two things: (You already tested the PSU so we don't need to worry about that)
Something is wrong with CPU (over heating or failed outright)
Something is wrong with mobo (failed)
if your CPU is overheating, then your mobo may simply be shutting down the system to avoid damage. reseat your heat sink to test but make sure you first get some thermal paste from the local computer store, you will have to wipe clean the contact surface of both the cpu and the heat sink and then reapply a small amount on the heat sink and spread it about.
if that still doesn't work, then we have either a failed CPU or a failed motherboard
your motherboard has several little capacitors on it, if one (or more) of them has burst
(capacitor plague) you can either try to solder on the appropriate replacement cap or buy a new mobo. This used to be very common on older computers but is rather rare these days assuming you don't have frequent power surges.
if you cannot see any bad caps on your motherboard the only way to isolate the issue is to put your CPU into another computer(make sure its compatible with the CPU) that is working and checking to see if it boots.
if it boots then your problem is the motherboard, buy a new mobo
if it does not boot then your problem is the CPU, buy a new CPU