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Old computer
If the computer is a few years old and/or has been used in a fairly dusty environment it's time to get out the vacuum cleaner, open up the computer case, and using the thin nozzle attachment carefully suck the accumulated dust, fluff and other gunk out of the system, especially the heatsinks, fans and power supply. Take care not to touch the motherboard or any of the cards or components while you're doing this. Sometimes this is all that's needed to get the show back on the road.
If this doesn't work the problem will almost certainly lie with the motherboard, graphics card, RAM modules or the CPU, so switch everything off, disconnect from the power source and open up the case again.
You might as well start with the easy things, so first check the state of the capacitors. These are unglamorous components looking a bit like small cylindrical batteries roughly a couple of centimetres high and a centimetre in diameter mounted vertically on the motherboard. If these have a brownish residue or corrosion on the outside, or if the tops are domed or at all convex as opposed to being perfectly flat, or if they're split or bulging in any way, that's where the trouble lies. Although capacitors can be replaced, unless you're highly skilled with a soldering iron it's advisable to simply replace the motherboard. If it's new enough to be still under warranty, contact the supplier and SME it.
Three capacitors showing residue/corrosion and convex tops. The motherboard was about 5 years old.
Further checks
If the capacitors look OK has the system clock been running slow lately? If it has you might have a failing CMOS battery. On many motherboards this will be a CR2032 3V Lithium Coin Battery sitting on top of the motherboard - very cheap and easy to replace. These batteries typically last at least five years and often up to ten years, but you never know, and if your computer is showing symptoms of a bad battery (slow running system clock and/or intermittent POST errors) it's worth replacing it just to make sure.
OK, you've ruled out capacitors and a failing CMOS battery, now what? Take out all auxiliary cards then carefully remove the graphics card. While you have the graphics card in your hand make sure the heatsink and fan (if it has one) are free from dust - give them a blast with an air-duster if necessary and be sure to give the graphics card slot and the card contacts a blast as well (if you don't have an air-duster carefully use a vacuum cleaner to clean out the heatsink and card slot).
Next carefully remove the memory modules. Again handle them with care and only at the edges. Use an old anti-static bag to put them on. Give the slots and the memory contacts a blast with the air-duster. Disconnect the motherboard power supply and give the connectors the air-duster treatment. Make sure that no screws or other foreign objects are rattling around on the motherboard. Remember that a clean computer is a happy computer!
When you're satisfied that all fans, heatsinks, sockets and the computer innards generally are free from fluff and dust, check round all the screws holding the motherboard in place to make sure nothing is over tightened. Now carefully replace everything except auxiliary cards (any card that isn't a graphics card), taking great care to ensure that everything is properly seated. For good measure try swapping the keyboard.
Put the lid back on the computer, reconnect, cross your fingers and switch on. With luck the computer should boot. If it doesn't and all you get is beeps, disconnect and open up the case once more. Try reinstalling the motherboard. While the motherboard is out of the case take the opportunity to clean up everything you missed the first time.
If the computer still refuses to boot you're now into a process of elimination. Switch off, disconnect and open up again. It's time to take out the graphics card and replace it with a known working one, does the computer boot up? If not take out the RAM modules and replace with known working module, if this doesn't do the trick do same with the CPU. If none of this does the trick replace the motherboard. As you've tried everything else, that should fix it.
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