In most cars the compressor is driven by a belt connected to the engine, and therefore the compressor runs at the speed of the engine. When the engine runs faster the compressor runs faster and provides better cooling.
However, there is a thermostat that when the evaporator gets to a preset temperature (usually just above freezing), the compressor clutch disconnects it from the engine, so it takes no load. So, although the compressor takes more power from the engine when the engine runs faster, this is compensated for because it runs less often and for shorter periods of time.
The faster you have your fan set, the more air is pumped through the evaporator. With the fan speed set to high the evaporator will warm up rapidly when the compressor stops, so the compressor will run more frequently, and will run longer. On pretty much all systems, if you switch the fan off, the compressor will also switch off completely - otherwise you'll turn the evaporator into a block of ice.
However, the temperature adjustment (not electronic controls) doesn't affect the AC - it's totally seperate and works on the air after its been through the AC. If you turn it up to warm (red), all that happens is the cold air from your AC is heated up by the heater. The AC still sucks in the same warm air from outside whatever the temperature you have dialed in, so the compressor takes the same load.
Some new cars have an electrically powered AC compressor, the power of these units doesn't vary with engine speed (but when running, will load down the engine through the alternator).