If you look at the shape of a typical aerofoil:
You see that the top is curved more than the bottom. The air flows over the top and bottom of the wing. Since the TOP flow has further to go, it has to move faster to "keep up". Bernoulli's law says that if the flow is faster, it's pressure must be lower.
To imagine this, think of a shoulder-to-shoulder crowd of people walking past an aerofoil shape (wing) standing on end. The people on the "top" side of the "wing" have further to walk, so they must speed up in order to keep up with the people on the "bottom" side of the "wing". Since they walk faster, they temporarily move further apart from each other (their density decreases).
Going back to airflow, decreased density means lower pressure. Therefore, there is a LOW PRESSURE area at the top of the wing. The wings are not pushed up by the airflow, they are SUCKED UP by the low pressure above the wing!
Ever take an air hose and blow the air straight up and suspend a ping-pong ball on the airstream? This works because the air flowing over the curve of the ball has to speed up as it flows around the ball, creating a low pressure area that pulls on the ball.
If the ball, say, tries to fall off to the left, the right side is exposed to more airflow, the pressure on the right side gets lower and the ball is "sucked" back into the center. The ball does "dance around" in the stream, but that's just due to turbulence in the airflow. But the ball is always kept inside the "column" of moving air due to Bernoulli's principle.
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