Since the mains supply is AC it has to be converted to DC to be useful in electronics.
This is what a power supply does.

In most cases the high voltage mains supply is too high for the electronic circuitry. 
It is therefore stepped down to a lower value by means of a Transformer.
The mains voltage can be stepped up where high DC voltages are required.

From the transformer the AC voltage is fed to a rectifier circuit consisting of one or more diodes.
The rectifier converts AC voltage to DC voltage.

This DC is not steady as from a battery. It is pulsating.
The pulsations are smoothed out by passing them through a smoothing circuit called a filter.
In its simplest form the filter is a capacitor and resistor.

Any remaining small variations can, if necessary, be removed by a regulator circuit which gives out a very steady voltage.
This regulator also removes any variations in the DC voltage output caused by the AC mains voltage changing in value.
Regulators are available in the form of Integrated Circuits with only three connections.