What is lift?

When a gas flows over an object, or when an object moves through a gas, the molecules of the gas are free to move about the object.
Because the molecules move, there is a velocity associated with the gas.
Within the gas, the velocity at different places near the object have different values.

Bernoulli's equation relates the local velocity to the pressure in a gas; so as the velocity increases the pressure must decrease. Adding up the  pressure variation times the area around the body determines an aerodynamic force on the body.
 
According to Newton's third low, a turning action of the flow will result a re-action (aerodynamic force) on the object. The integrated velocity variation around the object produces a net turning of the gas flow. Adding up the velocity variation around the object also determines an aerodynamic force on the body.

Integrating the effects of either the pressure or the velocity determines the total aerodynamic force on an object.

Lift is a component of this aerodynamic force.

Air velocity & pressure

Air has mass and moving air has momentum. If the air parcel follows a curved path, there must be a net force on it, as required by Newton's laws.

Using the law that total energy cannot change, whenever a given parcel of air increases its velocity, it must decrease its pressure, and vice versa.
This relationship is called Bernoulli's principle.

At high angles of attack a wing is extremely effective at speeding up the air above the wing and retarding the air below the wing. The maximum local velocity above the wing can be more than twice the free-stream velocity. This creates a negative pressure of more than 3 times  the positive pressure under the wing.

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Down Wash

According to Newton's 3rd law, the downward movement is the primary mechanism of lift, since the lift of a wing is proportional to the change in momentum of the air it is diverting down.

 

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Source: "Lift doesn't suck" by Roger Long.

Circulation

The whole system of motion is much like a wave in the water. The circulation of air around a wing is like a wave that moves with it, similar to the wake that moves along with a boat.

Air is pushed forward under the wing and accelerated backward over the wing . Also, air tries to move from high to low pressure. By the time the wing arrives, the air has established a pretty good up-wash ahead of the wing and, by the time the wing has passed, all the air is lower than it was before. Each particle of air is moved slightly up and down. The particles in the air mass are all bumping into each other, and each is affected by the ones next to it. The combination of these particles pattern and movements is called circulation.