One of the scariest things that can happen to your Mac, short of actual flames and smoke billowing from the case, is a failure to turn on at all. You press the power button and nothing happens--no startup sound, no light, nothing. If this happens, you can check several things before hauling your Mac to the nearest Apple Store for repair--as often as not, this seemingly difficult problem might have a simple solution.
First, make sure that your Mac is getting juice. To do this, you may need to trace the entire flow of electricity to your Mac. Check your Mac's power cord to ensure it is firmly seated where it connects to the computer as well as where it plugs into the wall. If it goes through an outlet strip or a UPS, make sure that's also connected and turned on. Also check that any surge protectors are still working--a power surge might have knocked them off.
You can confirm that an outlet is good by plugging in something else, such as a light. (If it doesn't work, check your fuses or circuit breakers.) If the outlet and all cable connections check out, make sure the power cord has no crimps, breaks, or other damage; a broken wire could be the culprit.
Faulty peripherals and broken buttons
Once you've established that your AC power path is good, it's time to look at your Mac itself. Unplug everything you can--not the power cord, obviously, or your mouse and keyboard if they're wired, and your monitor if it's not built in, but disconnect everything else and try pressing the power button again. If your Mac turns on, you know that one of your peripherals was at fault.
If your Mac doesn't turn on, it's worth trying to reset your Mac's SMC (System Management Controller), a chip that manages a number of hardware functions--including the operation of the power button. Directions vary by Mac model; see Apple's instructions for details.
If you have a Mac laptop, its battery should last through most power outages (from whatever cause), so you may not notice that you have a power-related problem until the battery runs out, at which point your Mac might simply appear to be dead. So try all the above tips, but also check your power adapter. If you have an AC cable attached to the adapter (as opposed to a plug going directly into the wall), make sure that cable is securely connected. If you have access to another AC adapter, switch to it briefly--that will tell you whether the original adapter is bad or whether it's something in your Mac itself.
When it's time for the pros
Still no luck? In that case, it's time for professional help. I've had Macs fail to turn on for reasons such as faulty logic board components and blown power supplies, but those sorts of things require heavy-duty intervention. An Apple Store or authorized repair center should be able to diagnose and fix the problem.