Assuming that the computer is doing absolutely nothing when the power button is pressed (no fans, etc...), it's probably very likely that the power supply has died.
EDIT: Before trying another power supply, I've had a few cases where ATX supplies with a power switch on the back of them, will seemingly act dead for no reason. After unplugging the supply from wall power for a few minutes with the power switch still ON, and plugging it back in, and it returns to normal again. You could try that before swapping out the supply.
If you happen to have another power supply laying around that is compatible with the motherboard, you many want to try it to see what the problem is. If it works using the new supply, then the old one is likely dead.
Or, if it still does not work, _odds_ are it's a problem with the motherboard, however, I did have one problem in the past where a power button on a cheap case managed to stick in the "pressed" position, even though the button did not look pressed. This, of course, caused it to do absolutely nothing.
If it does not turn on with the new supply, also be check for any grounded connections inside the case. Any wire that carries power should not be in direct contact with the case chassis (the wire itself touching does not matter, I mean if it's actually making _contact_, such as if a spot on the wire was rubbed down to the copper and touching the case), an example would be (I've done this before), a 3-pin fan extension cable laying on the bottom of the case, thus causing a short circuit, thus causing the Power Supply to enter 'panic' mode and shut down. They will also not power on in this condition.
