This principle states that if a fluid or air moves across a surface faster then the surrounding air, then the pressure for that area will be decreased.

Imagine a cross section of a wing:

/ ----___

Notice how the top of the wing is more curved (excuse my bad illustration). The wind going below the wing goes normally, but the wind trailing over the top follows the curve and meets up with the wind beneath. Since the speed is different, it creates this cool mini-vortex that leads to a lower pressure. The result is the pressure is negative above the wing, and the wing moves up to fill the vacuum.

A common misperception is that the wind molecules above and below the wing are matched up at the end. Nope, fortunately, because then the difference in speed would not give rise to a sufficent pressure difference. In fact, because of the sharp trailing end of the wing, it leaves behind a starting vortex, and correspondingly there is circulation about the wing, ie the speed of the air below it minus that above it is negative.

Bernoulli's principle states that the faster traveling wind above the wing creates a low pressure area, pulling the wing up in the air.

Some cars, especially racecars use this principle. Most drag racers usee this upside-down, so that the horizontal fin on the back of the car actually pulls them downwards towards the road.

Many thanks to Krimson for the Physics lesson