If you have an air conditioning unit that the outside condensing fan does not shut off, there is a few common things to look for.
The three most common problems are the thermostat, thermostat wiring, and Contactor.
While there are other things that can cause the same problem such as a bad circuit board the above are the most common to go bad.
The Contactor is the usual problem with them wearing out after many years of use.
Be sure to shut off power to a unit before examining any components on an air conditioning unit.
The Contactor is what turns power to a unit on and off and is controlled by the thermostat.
Low voltage from the thermostat creates a magnetic field which pulls the Contactor leads down letting power flow into a unit.
A bad Contactor will not pop back up a let power flow into a unit and not allow it to shut off.
Replacing a Contactor is not difficult but it does have high voltage going to it and into the A/C unit.
Unless you have electrical experience call a professional if you think the Contactor is bad.
Thermostat wiring can also short out and cause the circuit to seem closed to the unit and not let it shut off.
There is a few ways to troubleshoot this including using a multi meter to see if two wires have shorted.
A trick many HVAC techs use is to disconnect the thermostat wires at the outside unit and see if the fan shuts off.
This can also help to see if the thermostat itself has failed and is telling the outside unit to stay on.
Troubleshooting the above components will no doubt depend on your skill level.
If you are an electrician and know how high and low voltage works together you will likely have no problems fixing a unit.
Of course always call a professional if in doubt as high voltage can not only harm you but potentially the unit when done wrong.