I am trying to make a third person game. When moving my character, I want its direction to be determined by the position of the camera, not the direction my avatar happens to be facing. How do I do this?
Change your motion actuators to global instead of local
Try clicking on the L on the right. also does your character rotate mid game or just move left, right, forwards, and backwards? If you set the motion to global, the movement would be always in the same direction even if your character rotates. So as long as your camera doesn't rotate, the movement should be always the same and therefore the same with respect to the camera.
Hope this helps :)
Since you'r making a 3rd person game, I recommend watching this video on how to make a FPS setup. This guy also has a really great series for BGE.
Hope that helps.