If you have a light on the back, that's probably the LED on the motherboard. This means there's an output from the power supply - but it may only be the 5V. This light is usually not affected by the on/off switch.
When you say 'it does nothing', do you mean no lights on the front panel, no fan noise and no disk noise?
If all three, then the switch is not operating - it's not mechanically connected to the power supply, and the electronics may have gone.
If you have a light on the front panel, the switch is working, but the fans and disks all operate on 12V, and so the 12V side of the power supply may be gone.
It's unlikely, but possible, that multiple components would fail at once, so the power supply is the most likely. It's a straightforward job to replace it.