If you're certain it's plugged into a power supply correctly that's switched on, the switch on the back of the power supply unit in the computer is also on, if there is such a switch (some supplies don't have one), the cable to the motherboard from the power supply box is still connected firmly (it's the huge connector with I think 2x10 wires coming out of it), and then the switch on the front of your PC still doesn't do anything, then it sounds like you'd need to replace the power supply unit in the PC.
If you notice anything else unusual with the power supply box when you connect it to the mains, then that probably confirms something is wrong with it and it needs replacing, at least for safety if anything. When my last power supply unit broke, it would make a really loud high pitch sound if I plugged it into the mains, then the pitch would change very much like a discharging capacitor when I removed power