How is the RAT on the Airbus A320 family manually deployed?
There is a pushbutton on the overhead panel in the hydraulics section to manually deploy the RAT:


5 RAT MAN ON pb
The flight crew may extend the RAT at any time by pressing the RAT MAN ON pushbutton.
Note: The RAT extends automatically if AC BUS 1 and AC BUS 2 are lost. (refer to 1.24.20).

(A320 FCOM - Hydraulics - Controls and Indicators)
There is another button on overhead panel in the EMER ELEC PWR section, which will also deploy the RAT and connect the emergency generator:


1 MAN ON pb (guarded)
AUTO: When the following conditions are met:

AC BUS 1 is not electrically supplied.
AC BUS 2 is not electrically supplied.
Aircraft speed is greater than 100 knots.

.

The RAT extends.
The blue hydraulics system drives the emergency generator.
As soon as the emergency generator electrical parameters are within tolerance and the landing gear is retracted, the emergency generator is connected to the aircraft network.

Pressed: This selects manual RAT extension. Emergency generator coupling occurs 3 seconds after the RAT supplies the emergency generator.

(A320 FCOM - Electrical - Controls and Indicators)
Here is the full RAT deployment and emergency generator logic from the FCOM: