Sorry for a bit flippant answer, but I was on phone and the long answer is, well, long. Going to try to make a somewhat summary of it.

EDIT: Wanted to add a link to NASA's page about it, but unfortunately the links changed and I can't find the old (ca. 2000) site material.

You have a bunch of responses with Bernoulli's principle, but they are incomplete.

Essentially, the lift is higher pressure of air on the underneath of the wing, creating a force pushing the wing away. Now, how do you get said increased pressure?

Bernoulli's law shows that air moving with higher speed will give you effective lower pressure compared to air moving slower, and this is main source of lift in subsonic regime.

Now the secret is "why is the air under the wing moving slower than above it?" and the answer is much more complex than high school level physics textbooks try to say.

When airflow leaves the airfoil and mixes again, it creates a vortex rotating in the same axis as wing. If you were looking at a plane flying to the left, the vortex would be rotating counterclockwise. This is called sometimes a "bootstrap vortex", because it's rotation induces a vortex around the wing (in the hypothetical viewing angle, it would rotate clockwise). When you combine the induced vortex movement with forward movement of the plane, you end up with air underneath the wing moving slower compared to aircraft, and the air above the wing moving faster, giving you the conditions for bernoulli's law to provide you with pressure differential necessary for generation of lift.

Differences in Angle of Attack both impact parameters of the bootstrap vortex, impact pressures on the wing (but less than people would think, at least in the range where you get usable lift instead of stalling), and most importantly decide the rotation direction of the bootstrap vortex - which is why symmetrical wings work so long as you have positive angle of attack - and why flat plane doesn't work (because it doesn't create a good bootstrap vortex, just random turbulence).

There are simple experiments you can perform to prove the correctness of Bernoulli’s principle. If the math cannot accurately describe completely what is happening, then it is our math that is lacking. Technically the air is less dense on top which generates lift. And yes planes do fall from the sky when they stall due large angles of attack which disrupt the laminar flow.

Extra 300 would be one example. Generally you use symmetric profiles for acrobatic planes - as all profiles have their own pros and cons, and symmetric ones increase capabilities in inverse flight (and acrobatics) while sacrificing low speed and straight flight performance.

I am sorry but I disagree. Air as a gas is a compressible fluid. You can compress air with your hand. I don't understand why this is hard to grasp.

Basically it comes down to what is "lift."

It is known winged planes can not fly in a vacuum.

Rather planes fly by some sort of reaction force between air molecules and the wing.

You seem to suggest the wing is pulled up from the top. How does that work? Do air molecules have little hooks that attach to the upper wing surface?

Obviously no.

So lift is just another name for the ordinary newtonian reaction force. It pushes from the bottom.

How is that possible? It is possible because there are more air molecules colliding with the bottom of the wing than with the top. I.e. pressure is higher beneath than above.

These collisions transfer kinetic energy from the compressed air to the wing.