Switches in motor car air conditioning system?

AC switch turns the AC system on. That is, it operates the clutch on the compressor allowing it to be turned by the belt from the engine. This is how air con works. You have a system full of gas, the compressor squashes it. It expands again in a radiator type device inside the car called the evaporator. The evaporator gets cold and air is blown through it which comes out cold (and 'drier').
So the AC switch turns the compressor on, so it can get cold.
The fan switch turns the cabin fans on. The fans that make air blow at you.
Normally the fans will work without AC on, but AC will not work without the fans on as it would cause damage. Cars are wired so you cannot run AC without fans on. It will automatically turn off if you turn the fans off.

You shouldn't need to turn your AC off before turning the car off if everything is working ok.
However it can do the system good to blow full fans for 20 seconds or so with the ac off after you park. Helps stop the system getting stinky.

The fan switch turns on your blower motor which will simply pump out either cool or warm air, depending on your temperature setting.

The AC switch activate the AC compressor which will supply cold air to your vehicle.

Neither switch must be in the off position at start up. However, it might be best not to have your AC engaged when turning on the ignition.