What Mattz98 meant was, please tell us specifications of your machine:
If a machine from one manufacturer, tell us manufacturer name and model number, plus such details from your monitor.
If it's one that was custom built, what motherboard (exact model number), what CPU, what kind of RAM and how much, exactly what hard drives and what optical drives, what Power Supply maker and model number, etc.
What Operating System? - might be Windows, but exactly which version? Or, if something else, specify.
It will almost certainly be the Power Supply or if not that then the Motherboard. The only way to find out is to put in another Power Supply and see what happens. If that still doesn't work the Motherboard is the next suspect on the list. CPU problems are very rare. Their failure rate is very low (unless messed with).
1) Did you check the MOBO battery?
A lot of ppl seem to forget there is a small battery on the motherboard supplying power to te CMOS of the BIOS, in some cases the battery goes dead and the settings are corrupted. (I would expect no bios error code (= bleeps))
Some brands of MOBO allow for auto-reset-to-defaults on power-on or will just load basic settings in wich case you will get a 'normal'boot with some errors like incorrect sysdate/time, unrecognised HDD'S, missing or apearing FDD's changes in USB ports and settings, loss of raid config... etc.
What to do?
Just buy a new one (should cost 2 or 3 euro's or 4 dollars) and CAREFULLY replace it, empty out the bios by flushing it and restart the computer, reset time and date and thingies like your raid setup to correct values and see if it works... If it's below all your cards and hardware it will take you some time, next time buy a better board.
2) Static electicity due to dust bridges could be a cause.
Take a vacuumcleaner and VERY CAREFULLY! empty the dust from your system (especially the back of the MOBO/ area's between memory and around the CPU.) I saved about 12 ppl from buying a new pc by just taking this route, very usefull in student dorms (-evil grin-)... Cleaning out a PSU itself by opening it is NOT advisable, and last sometimes a small brush can do wonders in place you cannot reach easily (just use a small painting brush). A good indicator for this problem is excessive dust in your system fans...
3) Check for the fuse of the PSU, if burned out: replace PSU or you should have a very good idea why it burned in the first place (lightning strike?) and replace the fuse.
4) Did you overclock it? Some mobo's allow for this. if so flush the bios and try if it does anything. sometimes ppl's hardware degrades and leaves the PC unbootable, (BUT i would expect some bios error code) (sometimes this in combination with point 2 (dust) would cause the degrade.
5) DEAD MEMORY OR VIDEO? ===> listen to bios errorcodes (use your mobo booklet) check (and replace ) loss of bank 0 can mean failure to start.
6) DEAD CPU? ===> no bios errorcodes (only relevant when CPU fan is dead and with (extreme) high (O-)clocks) or when PC is in the sun whole day, especially with black casings. Hardware needs to be hotter than melting point of solder in the first place to get serious hardware dmg. BUT IN ALL SITUATIONS Check for temp. any temps above 65-70 degrees celsius are dangerous to the long term (and in case of cheap hardware short term) life of your rig. note that some laptops do not vent when used on lap and will fry themselves easily this way. Also NEVER EVER run a PC in a closed cabinet without ventilation as it gets very hot....
Have fun! the first 3-4 options don't cost a lot, last 2 maybe 3 can be very costly depending on the system you own. In all cases avoid static discharge (especially when using the vacuumcleaner as the bridges are a direct effect of static electricity and u probably use the metal pipe.)
(Edit: added point 2 as I always tend to forget to write it down for ppl, also I tried to remove a lot of typo's in my English hope you all can understand)
Past the edit:
DID you add a new piece of hardware recently? is it connected ok? does you PSU have enough watts available (some new videocards use more than my complete 2P server system (425watt on a 700watt PSU just for the record.) Alot of (budget-)computers have a 350-400watt PSU if you buy a Nvidia 400+ it will use ALL this power on it's own so your PC won't get a power-good status and boot, if it does boot like on a 450-525wattwatt it'll probably run very hot and burn out.
If you did replace or install something tihis would have been vital info and you should just replace your PSU with a (vastly) more powerfull one