The power supply is the component most likely to fail in a computer. Does the power supply's fan come on? That'd be the fan closest to where the power cord plugs in, btw.
Note: this requires opening the computer. When working on a computer, try to keep a hand on a metal part of the case or the power supply, or at least touch it before sticking your hand inside the computer.
If not, bad sign (or good sign, actually, since this indicates a high probability it's the power supply that died, and power supplies are cheap and relatively easy to replace). Open up the computer and unplug all power connectors inside except the one going to the motherboard (that'd be the biggest one). If your hard drives are SATA (red cable) unplug the red SATA cables too. Try to power on again and see if anything happens.
If the power supply fan does come on, open up the computer and see if anything else is coming on. CPU fan going? How about the video card's fan? Case fans? Is anything making any noise, lighting up, or anything at all? Be sure not to touch anything while the computer is on.