Yes, running your car while having the air conditioning turned on does burn or use more gas then if you had the A/C turned off.
Using the air conditioning in a vehicle turns on the compressor which makes the engine work harder to run this component of the car along with the other parts the engine is already running. The harder the engine works the more gasoline or fuel that it needs and uses.
However if you are comparing using the AC on the car to rolling down your windows there may not be a difference in the amount of gas used. Rolling down the windows and driving puts drag on the car and again makes the engine work harder to overcome the drag coefficient and thus uses more gas than if the windows were in the up position.
In all the studies I have read or seen they say the amount of extra gas used for running the air or rolling down the windows is about the same but that most people agree that it is more comfortable to have the air conditioning turned on then have the windows down in the heat of the summer.
If the weather permits you to neither turn on the air conditioning nor roll down the windows in your vehicle then you should get better gas mileage then in the warmer months when you will feel the need to turn on the A/C or roll down the windows to let the wind cool you down.