I own 2006 corolla. While driving I have noticed that the heater/AC fan speed changes significantly when you accelerate/decelerate. Headlights illunication changes as well. What could be the problem.

Could be the voltage regulator in the car. The regulator is connected to the alternator which controls the voltage to the cars electrical components and there could be a problem with the regulator. It's not uncommon for components to change speed or dim when you accelerate. Try to take note of when it is happening. Is it cold, are there a lot of devices being used in the car that could be draining the voltage (window defroster is a big one,) things of that nature. It would be very easy and quick for your local dealer to determine the specific problem. Hope that helps, Colin

Colin is right, but sometimes this is a simple simple problem.....you may have a bad battery. Your car is an 06, so this shouldnt be the case, but your battery may be defective. If its not the battery, then see Colins advice above.

The voltage regulator is most likely the problem. I had the same thing happen on a 99 Hyundai and had to replace the alternator twice. It would sometime peak so hard it blew out my headlights. Nowdays the voltage regulator is in the alternator instead of a separate part.