For the layman:
Visualize the typical wing with the front (Left) slightly raised:
'\' but only slightly.
More like this
' - .
It is moving left "
The air moves past wing to the right
'air --> ' - . wing
Look at the bottom. It is moving toward the air. It pushes on the air just like you push on the water in a pool when you try to run. You can feel the pressure you cause by pushing when moving toward the water. Same with the wing.
So, there is a little more pressure under the wing.
Above the wing, the air follows the curve. (BTW, this is NOT Coanda. It is a little similar, but it is not truly Coanda).
Don't worry about why we just know it follows the curve path over the wing. Also, it is the PATH of the air that is important, not simply the wing's shape. CURVED FLOW PATH.
This air is going around a curve just like you on a playground merry-go-round. You try to go straight but are pulled in a curve, so YOU are pulling away from the center of the curve.
On the wing, the air tries to pull away from this curved path which lowers the pressure at the surface.
[ NO Bernoulli. Fast air does not cause pressure to decrease. The fact is that higher pressure behind that air pushes it faster, so THAT is why it is moving faster when it gets to the low pressure - pressure behind is pushing it faster.]
SO, there is a little less pressure at the upper surface.
That's lift... a pressure difference top to bottom: More under pushing up than what's pushing down from above. The result is a 'net' upward push.
In addition, The pressures created BY the relative MOTION (of wing and air) ALSO cause all the other air movement around the wing as it passes by. I won't go into detail here, but see my references below.
It is those same pressures that also push a bunch of air down which satisfies Newton's Third Law that other people like to say "caused" lift.
Movement, pushes air around, causing pressure changes.
Fortunately, these changes push the wing up AND some air down.
Please, Lemme' know if this helps.
Some other answers address some of this but are pretty complicated.
P.S. Bernoulli does not teach us that fast-moving air has lower pressure. This is bad science repeated by way too many people.
Here are my full explanations: