Yes and no. And the full answer is more interesting than most people expect. The AC compressor is connected to the engine with the help of a belt, which means AC works only if the engine is running. When you switch on the AC, the compressor loads the engine significantly, and in order to compensate for the increase in load, the ECU orders the fuel injectors to spray more fuel, hence there is an increase in fuel consumption.
The compressor can require as much as 4 horsepower to power the AC system, which doesn't seem like much in normal driving conditions, but becomes an issue if you have a small engine or if you're towing a large load or accelerating fast. The smaller the engine in the car, the higher the fuel economy hit will be, as well as a noticeable loss of performance.
Here is where it gets more complicated though. The correct answer is really "it depends," because if you don't run the AC, are you driving with the windows down? Driving with the windows down creates turbulence and drag, and that uses more gas too, especially at highway speeds. At low speeds, around 50 km/h or less, there is not too much wind resistance, so having the AC on will use more fuel than simply opening a window. But once travelling at highway speeds, the wind resistance increases exponentially, and having a window open creates a lot more wind resistance than at lower speeds. At that point the extra wind resistance has a much more noticeable effect on economy, and it becomes more efficient to have the AC on.
In cars built in the last 15 years or so, the AC compressors are much more efficient than older cars. Around town, there will still be a little drop in mpg, but out on a steady highway drive, the aerodynamic drag of having open windows will cause a greater drop in mpg than having the AC on. Modern AC compressors are designed to get the system up to the correct pressure, then disengage so as not to waste fuel, so the pump will be running less often than most people assume. Keeping the windows closed when using air conditioning also reduces fuel consumption further.
The bottom line is that yes, the AC uses more fuel than driving with everything off and windows up. But if the choice is between AC with windows up versus no AC with windows down at highway speed, the AC is actually the more fuel efficient option. Keeping the AC off and the windows rolled up is most efficient, but since modern ACs use such little additional gas, most find the AC worth a slight decrease in range.
