Odds are, it is the power supply; but, to verify that we need some info and to try a few tests.
Unplug the AC power cord from the back of the PC and wait one minute to lett the standby voltage drain completely. Watch the led's on the keyboard while you plug it back in. All three should turn on for about 1/2 second and then go out. If that happens, try turning the PC on.
What is happening is that, when power is first applied, the BIOS tests several core parts of the system and then initializes the keyboard which is when the lights go out.
Because modern PC's supply a standby voltage (5 volts) 24/7 and that is a very common failure, the normal shutdown causes it to fail and then you have no power to turn it back on.
Before deciding it is the power supply, there is one other thing to check. Take the side off the PC and carefully inspect all of the capacitors which look like small soup cans with X's, Y's, or K's scratched in the tops. There will be 15 or so immediately next to the CPU; but, also look at the rest on the systemboard. If any of them look puffy or have had the scratched lines in the top rupture (they usually ooze rusty brownish stuff), your problem is bad capacitors. Look here: http://en.wikipedia.org/wiki/Capacitor_plague
Tell me which problem you have and what the make and model of the system is so I can help you get the right part, OK?