Diehardbiker mentioned exact thing but if you work on OBD-II vehicles, start with a scanner tool first. Why? Computer stored the codes of CTS (Coolant Temperature Sensor) and Thermostat sensor (new to me, found in European cars like BMWs) even the MIL is not turn light on or previous mechanic/DIYer cleared the codes. If no codes, then go over the front engine compartment to inspect the fluid levels like coolant, oil etc. If coolant in radiator is low level or empty, this is problem for A/C operation because the computer will shut the A/C system when the engine reach over 225 degree F or more. Find nothing wrong. Then take a close look at the belt(s) for sign of loose and A/C clutch for sign of yellow/rusty color. Why? Loose belt cause the compressor slippage another one is compressor speed sensor (NOT ALL the cars have one) that computer will read the engine and compressor speed for compare to match speed if the compressor cant pick up speed then computer will shut the compressor down cuz the computer think it is loose belt or seized compressor. Smart computer.
If you dont have a scanner tool, you can check the electric cooling fan motors
(radiator/condenser fan motors) to see if fan or both fans runs with A/C control panel is turn on then check the clutch like Diehardbike mentioned.
If the clutch engaged and runs then put your hand on Low side line at the firewall side not to front condenser, and feel to see if it is very cold and sweat that mean A/C operation is ok if no blow or no cold air inside the car you have HVAC problem inside like broken blend door or bad blower motor or clogged cabin air filter or debris clogged in the evaporator core.
For old cars are much easy to diagnose or fix. Pm me if you have any questions