Old computers (AT type power supply) had a hard on/off switch - the switch on the front of the PC directly controlled access to the 120VAC power coming in from the wall. With that type, when your PC was off, it consumed no power, since it was no longer electrically connected.
With ATX, you now had soft-off. The motherboard would always receive power from the power supply. Pressing the power button on the front of the computer case now connected a low-voltage signal to a receiver, which would cause the power supply and system to fully power up.
This constant power to the motherboard allowed for other things, such as the network boot feature or timer-based power-on that Paperwastage mentioned. With these things, a chip on the motherboard could be running while the system was not actively running. That chip could then instruct the rest of the system to power up.
More recently, emphasis has been placed on power supply efficiency during low-power modes, so that it's not burning >10W just to provide power to a few small chips that can run on less than a few milliwatts each when idle.