The amazing science of flight is largely governed by Newton's laws.
Consider a wing cross-section:
Air hits it at a certain speed. However, the shape of the wing forces air to rush over it and under it at different rates. The top curve creates a partial vacuum - a region "missing" a bit of air. So, the pressure (force/area) on top of the wing can become less than the pressure below. If the numbers are right, and the resulting force below the wing is greater than the weight of the plane, the plane can lift.
This is often expressed as the Bernoulli Principle:
Pressure in a moving stream of fluid (such as air) is less than the pressure of the surrounding fluid.
The image above shows another way to think of flight - imagine the wing first shown, but slightly inclined upward (to exacerbate the effect). There is a downward deflection of air. The reaction force from the air below provides lift and the lift is proportional to the force on the wing.
In practice, it works out (in general) to be:
Lift = 0.3 p v^2 A
where p is the density of air, v (squared) is the speed of the plane, and A is the effective area. Note that the lift is proportional to the speed squared - so, the faster the plane goes, the (much) easier it is to take flight.
Some related animation:
http://physics.stackexchange.com/questions/13030/why-does-the-air-flow-faster-over-the-top-of-an-airfoil
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