What is the popular theory?

Lift is usually explained with the equal transit times theory, which was abandoned a century ago and which is flat out wrong. You find this theory in schoolbooks, though it has been known for a long time it's a false simplification motivated with a lie. For instance, how would inverted flight be possible? What would we need an angle of attack for, shouldn't that worsen the lift? The theory assumes that the airflows below and above the wing move a different distance in the same time, causing a pressure difference by the Bernoulli effect and then lift. The simple rebuttal is: how does the air molecule below know it has a date with the air molecule above the wing?

Why the popular theory doesn't work?

This theory has many weak points. For instance, it does not agree with the reality: the airflows do not meet. Reality is always right in natural science. However, the airflow above does flow faster. Could Mr. Bernoulli explain it alone? Eberhardt and Anderson (2001) have calculated this:

Take the case of a Cessna 172, which is popular, high-winged, four-seat airplane. The wings must lift 2300 lb (1045 kg) at its maximum flying weight. The path length for the air over the top of the wing is only about 1.5% greater than under the wing. Using the Popular Description of lift, the wing would develop only about 2% of the needed lift at 65 mph (104 km/h), which is "slow flight" for this airplane. In fact, the calculations say that the minimum speed for this wing to develop sufficient lift is over 400 mph (640 km/h). If one works the problem the other way and asks what the difference in path length would have to be for the Popular Description to account for lift in slow flight, the answer would be 50%. The thickness of the wing would be almost the same as the chord length.

So, how does it work?

Cessnas don't fly 640 km/h. The equal transit times theory is theoretically impossible and experimentally insufficient, but what really proves it false is that a wing does work, force times distance. We have the gravity, which constantly accelerates the airplane downwards. Free air is not a support, no matter at which pressure it is at, so the airplane should accelerate towards the ground. A force is a rate of change of momentum. The weight of the airplane is a force, so what do we do to prevent the plane from falling? Change the momentum of air instead of the plane! We conserve momentum, it's only air that's falling down fast, not the plane.

How does the wing do this? The reason is the Coanda effect. Air, no matter how thin it is, is a fluid, a flowing substance. Fluids tend to follow curved surfaces - this is the Coanda effect. Look at the wing: the convex top surface of the wing is curved such that it points slightly downwards. In practice, this angle is 5-15 degrees. Incoming airflow flows along the top and is expelled downwards - a change of momentum. By Newton III, when the wing forces air down \/, the air forces the wing up /\. This force is called lift. In aviation terms, the flow of air downwards is called downwash.

A typical Cessna forces downwards 6 ton/s of air to counteract its weight of one ton. For incoming air, the top surface of the wing generates a "scoop" that forces air downwards. (Eberhardt and Anderson 2001).

Lift is also generated from the airflow below the wing. The surface below wing at an angle - angle of attack - deflects air downwards. The total impulse on the wing points upwards at an angle to the vertical. However, this force is small compared to the force from the air above the wing. The top is the critical surface, which explains why extra fuel tanks, bombs, etc. are placed below, not above the wing.

Show me an example!

You can demonstrate Coanda lift with a thin stream from a tap/faucet and a spoon, which functions as a wing. Suspend the spoon such that convex part is against the stream - call this direction "up". Let the spoon lightly touch the stream. The stream immediately sucks the spoon "upwards", leaving it hanging at an angle. You can see how the spoon forces the water "downwards" and the water forces the spoon "upwards".

Why there's a Coanda effect?

Why does air follow the curve of the wing? This is the point where somebody messed up with the Bernoulli effect. Now, imagine a wing at a good angle of attack. A fluid flows above the wing. The fluid just behind the top surface is sucked in to this flow. This would generate a vacuum above the wing if nothing else happened. You can think of the vorticity a vacuum just above the wing would induce in the incoming airflow. This "potentially vacuum" area is filled by fluid from the incoming flow. There! The fluid accelerates downwards and flows along the curve of the top surface. Similarly, in the area just below the wing, air would accumulate and generate infinite pressure, so it flows downwards, changing its momentum and giving lift. So, we do have a difference in pressure, which generates force.

We see the Coanda effect and understand the whole picture best with it. For the air, communication of force is possible only with pressure and this can be misunderstood as the only thing that's happening. Most popular explanations in textbooks ignore the propulsion of air downwards. That's analogous to kids' Computer Science books explaining the operation of a computer with zeros and ones and not mentioning digital logic gates. Yes, that's what happens inside the system, but that's not why the system works.

For a more exhaustive explanation and more explained phenomena, such as wing efficiency, ground effect and wing vortices, see Scott Eberhardt's and David Anderson's webpage on their book Understanding Flight.
More myths at William J. Beaty's Science Myths in K-6 Textbooks and Popular culture.
Thanks to Ken Fuller. Grammar corrected by Shr00m.

Replying to weasello above: "In any case, time will eventually show that all of these crazy theories were all wrong and some grand unified theory will explain everything." You are using the "just a theory" argument. It is a problem of the precise definition of "theory" and "true" in the English language. We are trying to explain, not prove a theory.

The "true" source of any force on an object suspended in free air can only be a difference in pressure. However, this does not give a clear picture. The Coanda effect gives. This isn't proof and isn't meant to be. Proof is actually applying the equations or making a simulation.

Brontosaurus says Interesting, but how does this explain inverted flight?
Inverted flight isn't different from conventional flight. Planes designed for inverted flight have symmetric wings, so that when turned upside down, the wing has the same shape for airflows as before. It's only dependent on the angle of attack at which the airflow meets the wing. Acrobatic airplanes have symmetric wings and generate lift up is the nose is up and down if the nose is down. This makes it easy to fly inverted: just point the nose at the same angle with respect to vertical.