Why air has weight, and that is the whole subject

Why every result in aerodynamics follows from air being a substance with mass.

Air has weight, a small but completely real mass in every lungful of it, and that single fact is the reason a curved wing can lift a loaded airliner, a spinning propeller blade can drag an aircraft forward, and a stalled wing can lose its grip on the air in an instant: every one of those is just a different way of pushing around a substance that has something in it to push.

The short version

It is easy to think of air as nothing, an absence rather than a substance, since it is invisible and offers so little resistance to a hand waved gently through it. But air is matter, made of the same kind of molecules as anything else, just spaced far apart and free to move, and matter has mass. Every mechanism already covered, the reaction force from a tilted wing, the pressure difference that holds an airliner up, the thrust a propeller blade generates, the collapse of lift in a stall, depends on air having that mass, because a substance with no mass could not be deflected, could not carry momentum, and could not push back on anything deflecting it.

The physics of a substance with mass

Newton's laws, the ones that explain why a wing rises when it deflects air down and why a propeller blade is pushed forward when it deflects air back, only work because there is something with mass being deflected. Force is mass multiplied by how quickly that mass's velocity is changing, and a wing or a propeller blade is constantly accelerating a stream of air, speeding some of it up, slowing some of it down, and above all changing its direction. None of that produces a reaction force unless the air being accelerated actually has mass to accelerate. The same fact explains why a balloon floats at all: floating is buoyancy, governed by Archimedes' principle, and that principle only produces an upward force because the air being displaced by the balloon has weight of its own to lose. A vacuum has nothing to deflect and nothing to displace, which is exactly why wings, propellers, and balloons alike stop working the instant the surrounding air runs out.

The bicycle comparison

Riding a bicycle at speed on a still day gives a direct, physical sense of air as a substance rather than an absence. The push felt against the chest and face is not imaginary and is not merely the sensation of movement, it is the measurable result of a rider's body accelerating a stream of air out of its way every second, and that air, having real mass, pushes back exactly as hard as it is pushed. Standing still, there is nothing to feel, since nothing is being accelerated, but the air was there all along with the same mass it has while cycling through it. What changes between standing and riding is not the air, it is how hard the air is being pushed, and therefore how hard it pushes back.

The one number worth remembering

Air is far denser than it feels. A cube of air a metre on each side, roughly the size of a large upright refrigerator, weighs about as much as a bag of sugar bought at a shop. A wing moving through the sky, or a propeller spinning in front of an engine, is not delicately nudging a wisp of nothing, it is shoving aside a genuinely heavy substance, a slab the size of a small room's worth of air every second for a mid-sized aircraft, and that is the raw material every pound of lift and every pound of thrust ultimately comes from.

What follows from this

Because lift and thrust both depend on how much mass of air a wing or propeller can get hold of and accelerate each second, thinner air changes everything even when nothing else about the aircraft or its speed changes. At altitude, where air is spread more thinly than it is near the ground, the same wing moving at the same speed deflects a lighter slab of air each second and produces less lift for it, and the same propeller spinning at the same rate pushes back a lighter slab of air and produces less thrust, which is why aircraft need to fly faster at altitude to make up for the thinner substance they are working with, and why engines built to breathe dense air near the ground struggle increasingly as that air runs thin.