Do phones have a power supply?
Power Supply is actually a term with some actual, albeit generic, significance and meaning. 

In handsets, the power supply is probably(I'm not an electrical engineer, or expert on device hardware) a component that regulates the power supplied to the device (keeping the voltage constant, for one thing), and manages the physical charging/discharging of the battery.

As far as what could have damaged it, it could be anything from dropping/shaking the device if there is a loose connection, to a short caused by water, to overheating. Unless you can find a connection that got loose, and the power supply is indeed fried, you'll need to get it repaired.

If you think there might have been any water damage, put the phone in rice for a day or three, it will dry out most liquid that may be in the device, and it might just work.