Power supplies convert one voltage into another to suit the power requirements of a device.
The most common case is conversion of 230V alternating current into a direct current, such as 12V, but there are several other kinds of voltage conversions.
Regulated AC adapters include rectifiers and voltage regulators, and deliver a very precise voltage, only deviating around 0.1 volts from the nominal voltage, regardless of input voltage and load (within limits).
Variable means that the ouput voltage can be selected with some type of control on the power supply itself.
Fixed means that the ouput voltage is completely fixed. It is usually regulated.