What really allows airplanes to fly? What aerodynamic effects actually contribute to producing the lift on an airplane?
I know there's a common belief that lift comes from the Bernoulli effect, where air moving over the wings is at reduced pressure because it's forced to travel further than air flowing under the wings. But I also know that this is wrong, or at best a minor contribution to the actual lift. The thing is, none of the many sources I've seen that discredit the Bernoulli effect explain what's actually going on, so I'm left wondering. Why do airplanes actually fly? Is this something that can be explained or summarized at a level appropriate for someone who isn't trained in fluid dynamics?
(Links to further reading for more detail would also be much appreciated)
 A: From Stick and Rudder by Wolfgang Langewiesche, page 9, published 1944:

The main fact of all heavier-than-air
  flight is this: the wing keeps the
  airplane up by pushing the air down.
It shoves the air down with its bottom
  surface, and it pulls the air down
  with its top surface; the latter
  action is the more important. But the
  really important thing to understand
  is that the wing, in whatever fashion,
  makes the air go down. In exerting a
  downward force upon the air, the wing
  receives an upward counterforce--by
  the same principle, known as Newton's
  law of action and reaction, which
  makes a gun recoil as it shoves the
  bullet out forward; and which makes
  the nozzle of a fire hose press
  backward heavily against the fireman
  as it shoots out a stream of water
  forward. Air is heavy; sea-level air
  weights about 2 pounds per cubic yard;
  thus, as your wings give a downward
  push to a cubic yard after cubic yard
  of that heavy stuff, they get upward
  reactions that are equally hefty.
That's what keeps an airplane up.
  Newton's law says that, if the wing
  pushes the air down, the air must push
  the wing up. It also puts the same
  thing the other way 'round: if the
  wing is to hold the airplane up in the
  fluid, ever-yielding air, it can do so
  only by pushing the air down. All the
  fancy physics of Bernoulli's Theorem,
  all the highbrow math of the
  circulation theory, all the diagrams
  showing the airflow on a wing--all
  that is only an elaboration and more
  detailed description of just how
  Newton's law fulfills itself--for
  instance, the rather interesting but
  (for the pilot) really quite useless
  observation that the wing does most of
  its downwashing work by suction, with
  its top surface. ...
Thus, if you will forget some of this
  excessive erudition, a wing becomes
  much easier to understand; it is in
  the last analysis nothing but an air
  deflector. It is an inclined plane,
  cleverly curved, to be sure, and
  elaborately streamlined, but still
  essentially an inclined plane. That's,
  after all, why that whole fascinating
  contraption of ours is called an
  air-plane.

A: I'm late to the party here and I think the top vote-getters (Sklivvz, niboz) have adequately answered it, but I'll give my two cents anyway:
There are several ways to explain how an airplane flies.  Some are more detailed than others, and unfortunately most popular explanation get it wrong. Here are some explanations that are useful, depending on the audience:


*

*The simplest explanation is that the wing pushes the air down and
according to Newton's third law the air exerts an equal but opposite
force up. The main way this happens is via the angle of attack, but
the shape of the wing also plays a part. This suffices for most
people, and should be the default explanation.

*A more detailed explanation would discuss the pressure difference
between the two sides of the wing - since lift is a mechanical force
it must be exerted on the surface of the wing and the only way air
can do that is through pressure.  So there must be a region of low
pressure on the top of the wing and higher pressure on the bottom. 
Where does this come from? It comes from the air changing direction
as it flows around the wing.  Whenever air changes direction and
follows a path that is curved there are pressure gradients with lower
pressure on the inside of the curve.

*A still more detailed explanation would be to examine the
Navier-Stakes equations and all the attendant math that goes with
them.  That's beyond the scope of this answer.
Holger Babinsky wrote a very readable paper called "How Do Wings Work?" that I'd recommend.  It covers the middle answer quite well (and refutes a lot of the nonsensical explanation that are unfortunately all too common). Knowing a bit of calculus is helpful, but I think the article is readable without it. See http://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf
A: Nib's answer is correct. The highly upvoted answer from Sklivvz starts promising but then throws in some incorrect statements:

Explanations showing a wing profile without an angle of attack are incorrect. Airplane wings are attached at an angle so they push the air down, and the airfoil shape lets them do so efficiently and in a stable configuration.
This incidence means that even when the airplane is at zero degrees, the wing is still at the 5 or 10 degree angle.

An asymmetric aerofoil creates lift at AoA zero. All fixed wing aircraft have asymmetrical aerofoils, only helicopters use symmetrical wing profiles in the rotor (due to these having no twisting moment). Fixed wing aircraft have wing twist: they have a positive angle of attack at the root, a negative AoA at the tip, and an average AoA as close to zero as possible, to minimise drag.
Indeed what makes the airplane fly is deflecting an airstream downwards. A flat plate can do this, and Bernoulli has no place in a flat plate. Subsonic aircraft don't use flat plates because they create a large amount of drag at angles of attack other than zero - in fact in a turbulent flow, even a flat plate at AoA zero creates more drag than a symmetrical wing profile such as NACA 0012.
A: Since you asked for an explanation appropriate to an non-specialized audience, maybe this will do: "A Physical Description of Flight; Revisited" by David Anderson & Scott Eberhardt. It is a revision of the earlier "A Physical Description of Flight" (HTML version).
A: Consider the velocity field of the particles in the air mass in a 2D projection of the X(forward) and Z(up) axes. For each particle, Integrate over area and time, to derive the center of air-mass momentum (p) before and after the passage of the airplane : dp/dt. (On a very calm morning, with no wind or turbulence, the center of air-mass and its momentum is stationary in Z(assume level un-accelerated flight), and equal to the True Airspeed in X pointing in the aft -X direction. Integrate over the area and you will find that the center and momentum of the particle and vector field has changed, with passage of the plane. This center of air mass and center of momentum will move forward(+X) and downward(-Z) relative to its original state. The equal and opposite momentum change with time dp/dt of the airplane is a force. We might label the -X component "drag" and the +Z component "lift"(careful: the airplane coordinate system is different from the stationary airmass). This is a dissipative system, so don't wait too long after the plane passes to record the vector field. We can observe this process in contrails on clear days when the high altitude air is cold and relatively moist. Sadly since we mostly view them from below with a projection along the Z, we miss the downward component of the momentum field. You can see this as a test pilot, flying as chase wing-man, in formation (projection in the Y-Z plane from behind or X-Z from the side).
 Expand this model to 3D to include lateral or Y axis flow and effects!
 I suggest this "p-dot"(dp/dt) of momentum-change explanation is better, than "pushing" or "pulling" the air downwards, because the later may confuse position and momentum in the view of the reader. This is also the first term(LHS) in the beautiful Euler-LaGrange equation, which would lead to an even more elegant analysis of this question!
As a new user, I will need to figure out how to attach the appropriate Figures and Equations to this post...-thanks
Note: The drag equation is really the ideal gas law, except density replaces m/V.
P/rho = R T : 
A: Essentially a fixed-wing aircraft flies because it moves through the air and has a fixed wing which is angled to the direction of airflow.  A component of the drag force acting on the wing acts in the direction (up) opposite to the direction (down) of the aircraft's weight force. 
An aeroplane wing acts like a weather vane responding to the relative flow of air.  The basic effect can be obtained with a stiff, flat plate and a source of forward motion such as a propeller, gravity or launch momentum (e.g. children's paper planes).  Refinements (such as aerofoil cross-sections) are introduced to mitigate the undesirable side-effects of flat plates (such as stalling).
No great argument with the other popular answers here but I will try to explain the basics of fixed wings in terms of Molecular Collisions.  The following is rather a simplified explanation (ignoring things such as temperature, density, viscosity, compressibility, shear, boundary layers, turbulence, vortices, form drag, wing roughness, stiffness, skin friction, stalling, transmission by chain reactions, force couples etc).
A thought experiment. You sit at the bottom of a deep, water-filled swimming pool.  You hold a table-tennis bat in one hand. Extend your arm and try to sweep the bat horizontally at constant speed through the water with the face of the bat firstly (a) vertical, then (b) horizontal, then (c) somewhere in between.  
In case (a) the bat face is vertical and there will be the greatest resistance to forward motion.  The resistance to forward motion can be explained by two broad effects.  
The first effect is because the water molecules colliding with and rebounding elastically from the front face of the bat do so slightly faster and more frequently (on average) than the water molecules hitting the rear face of the bat. This is a simple consequence of the bat moving in the forward direction and the conservation of linear momentum in elastic collisions (think billiard balls hitting a large, massive, stiff, smooth, flat steel mirror). Each collision causes a change in the velocity of the bat.  Because the frontal collisions are on average faster and more frequent than the rear collisions the net effect will be to reduce the forward velocity of the bat.  In order to keep the bat moving at constant speed through the water you will need to expend muscular energy doing work against the resistance.
The second effect follows from the first effect.  The molecules colliding with the front of the bat will be swept forwards causing an increase in pressure (a ram effect).  This increase in pressure will act to further increase the air molecule velocities and rates of collision at the front face of the bat.  The zone of increased pressure will grow in size ahead of the bat.  Over time the continued growth of the high pressure zone will be offset by lateral diffusion of kinetic energy (high velocity molecules donating some of their velocity to surrounding slower-moving molecules by elastic collisions) and by mass flow of molecules past the edges of the bat to the lower pressure areas to the rear of the bat.
In case (b) the bat face is horizontal and the bat slides through the water with relatively little resistance.  
In case (c) the bat face is tilted.  The magnitude of the resistance depends on the angle of the bat face relative to the direction of motion.  The resistance is greater when the bat face is near-vertical (steep angle of attack) compared to when the bat face is near-horizontal (shallow angle of attack). The resistance magnitude depends on the apparent cross-sectional area of the bat facing in the direction of motion.  At shallower angle of attack fewer molecules impact the bat face, average angle of incidence of particles arriving at the bat face is greater causing reduced momentum exchange and there is less upstream pressure build up because it is easier (less obstruction) for molecules to escape the high pressure zone by flowing past the bat.
When the bat face is tilted upwards the net force on the bat is directed not backwards horizontally as in cases (a) and (b) but perpendicularly to the bat face (part backwards and part upwards).  This can be explained by the geometry of molecular collisions at a flat surface moving through a stationary fluid.  
A classical aerodynamicist might describe the face-perpendicular accelerations as combining components of both drag (backwards) and lift (upwards).  If you tilt the bat so that the leading edge is tilted downwards then the net direction  of the resistance to bat motion will be part backwards (drag) and part downwards ("negative lift").  Unqualified use of the term "lift" may cause confusion.  It may be better to refer to components of wing-induced drag operating in specific directions (e.g. upwards, perpendicular to main airflow, perpendicular to wing surface, perpendicular to horizontal plane of the aircraft).    
You can get a good feeling for the basic wing-induced drag effect by holding your hand, flat with fingers together, out of the window of an automobile when it is traveling fast (say 50 mph) and tilting your palm up and down and noting the forces which you feel when trying to keep your hand in the same position. (Probably best not to try a table tennis bat on public roads!).
A: A short summary of the paper mentioned in another answer and another good site.
Basically planes fly because they push enough air downwards and receive an upwards lift thanks to Newton's third law.
They do so in a variety of manners, but the most significant contributions are:


*

*The angle of attack of the wings, which uses drag to push the air down. This is typical during take off (think of airplanes going upwards with the nose up) and landing (flaps). This is also how planes fly upside down.

*The asymmetrical shape of the wings that directs the air passing over them downwards instead of straight behind. This allows planes to fly level to the ground without having a permanent angle on the wings.


Explanations showing a wing profile without an angle of attack are incorrect. Airplane wings are attached at an angle so they push the air down, and the airfoil shape lets them do so efficiently and in a stable configuration. 

This incidence means that even when the airplane is at zero degrees, the wing is still at the 5 or 10 degree angle.

-- What is the most common degree for the angle of attack in 747's, 757's, and 767's


Any object with an angle of attack in a moving fluid, such as a flat plate, a building, or the deck of a bridge, will generate an aerodynamic force (called lift) perpendicular to the flow. Airfoils are more efficient lifting shapes, able to generate more lift (up to a point), and to generate lift with less drag.

--Airfoil
A: Wings provide lift because they direct air downwards.
They direct air downwards in two ways. In part, the bottom of the wing slopes downward a bit and just pushes the air down as it moves forward through the air. But this is a small effect. The top of the wing is more important.
The top of the wing pulls the air down partially by providing a ramp. The rear portion of the top of the wing slopes down to a sharp trailing edge. The air, which is under pressure from the miles of air above it, follows that slope down the wing, and continues downward after the wing has passed. 
But there is more to it than that. As the wing drives forward, the air that is deflected upward by the leading edge ends up being pinched between the layers of air above and the bulging top of the wing. That pinching makes the air speed up, not so differently from the way pinching a wet watermelon seed can send it flying. The inertia of the air that is farther from the wing forces the air that is closer to the wing to hug the wing's top surface, reaching the trailing edge much sooner than the corresponding molecules that headed along the bottom. 
The asymmetry, of course, is key here. The bottom of the wing is more nearly parallel to the path of the air, with a bit of a downward slope all the way to the back, so it doesn't have the same pinching effect. (The asymmetry doesn't have to be in the shape of the wing. It can all be in the angle of attack. You are still creating a scenario where the air is pinched more on one side than the other.)
Of course there is no clear boundary between the layers of air that are doing the pinching and the air that is being pinched. But still, the force of the wing is felt most strongly by the air that is closest, and so that layer is accelerated the most. Each bit of air pinches the air below and is pinched against the air above, to a decreasing degree, until the effect is no longer noticeable quite some distance above the wing.
All this accelerated air is subject to the Bernoulli effect. Because it has been accelerated, its downward pressure on the wing is less than the upward pressure of the air below, and also the upward pressure on the air above is less than the ambient pressure. This causes even more air to move downward than otherwise would do so. Unless I am mistaken this is an important part of the downward deflection of the air.
The myth, then, is not that the Bernoulli effect is important. The myth is that there is an equal-time principle that is the reason the air atop the wing moves faster.
But the explanation is still incomplete because the Bernoulli principle itself is not obvious. The principle is often explained in terms of the low pressure causing the acceleration -- if you create an area of low pressure, air will indeed accelerate towards it. But if you blow into a tube with a construction, the decrease in pressure at the constriction will try to constrict it more. The upstream pressure from your lungs really is causing the decrease in pressure; it is not just the lower pressure that is causing the air to flow.
The way that increased pressure in your lungs can cause decreased pressure at the constriction is that your lungs give the air momentum. When the air finally leaves the tube that momentum is absorbed by the surrounding air, pushing it back like a mob pushing into a standing crowd. That momentum keeps some of the back pressure from being felt by the moving air in the tube. The higher the speed, the less momentum density, and the less back pressure. 
In fact, in a steady state, inviscid, incompressible model, the question of what causes what becomes almost meaningless. The air speeds up because there is lower pressure in front, and there is lower pressure in front because of the velocity of the air. But in the case of an airplane, my understanding is that that the thrust of the engines is causing the acceleration of the air by more than just letting the downward sloping top of the wing recede from it. Even at high subsonic speeds where the air can no longer be treated as incompressible, the qualitative phenomenon that greater velocity leads to reduced pressure still applies. Calculating the effect just becomes more complicated.
Frequently, Bernoulli's principle is derived using conservation of energy along streamlines. I think my qualitative explanation using momentum is consistent with that. 
The principle of lift is often explained using circulation. Again, I think that is just a different way of describing the same process. The different velocities along the top and the bottom constitute a net circulation.
Note: See "Why does the air flow faster over the top of an airfoil?" for additional answers to that part of the question of lift.
A: Without going into the excellent and detailed mechanics explaining reaction lift that others have provided for this answer, I just want to say that contrary to popular belief/high school physics textbooks, airplanes do not fly solely on account of Bernoulli's principle. According to Walter Lewin's excellent "For the Love of Physics":
"Bernoulli's principle accounts for 20% of an airplane's lift, the rest is provided by reaction lift."
Walter Lewin also poses an insightful question if planes really fly due to the equal transit theory and Bernoulli's principle (they do not!). 
"...then how do planes fly upside down?"
A: Because of the obstruction of the wing, the air has to go around the wing, so the air pressure at the bottom of the wing is increased because the air at the bottom of the wing is compressed to go around the wing, and the air at the top of the wing is stretched around the wing, so the air pressure at the top of the wing decreases. So there's a pressure difference, and then there's a lift.
Note: The bottom of the wing is windward, so the air is compressed, the pressure is high, and the top of the wing is leeward, so the air is stretched and the pressure is low.
So lift cannot be explained by Bernoulli's theorem. Because Bernoulli's theorem does not consider the compression and stretching of fluid.

The following is a detailed explanation:
For example, at the top of the wing, the direction of air velocity at point A is the direction of the blue arrow. Because the blue arrow is inclined (note the angle between the blue arrow and the blue normal in the picture), the blue arrow tends to be far away from the wing along the normal direction at the top of the wing, so the air pressure at the top of the wing is stretched, so the air pressure at the top of the wing decreases, so there is a pressure difference (pressure gradient). This pressure difference changes the direction of air velocity, so the direction of air velocity at point B is the direction of the red arrow, and the red arrow is also inclined.... So the direction of air velocity will continue to change along the top of the wing.
It should be noted that this pressure difference not only changes the velocity direction of the air on the top of the wing, but also generates the lift of the wing.

A: Fluid interactions with solid bodies depend on the fluid properties and the geometry of the object.  In the case of an aeroplane, we have air as our fluid and an aerofoil geometry.  The aerofoil geometry is designed on purpose to force fluid under it preferentially to above it.  This results in a pressure difference, which then leads to a buoyancy force that accelerates the wing according to Newton's second law (lift).  Bernoulli's law is relevant for calculating the fluid problem.
So, to achieve flight, all you need is some well-designed aerofoils and some way of imparting an initial velocity.  To keep flying you need to keep your speed high and to keep flying stably you need a well-designed aircraft with the centre of mass, centre of thrust and centre of lift being in the same position.
A: If there is no low pressure (negative pressure) at the top of the wing, will the airflow move downward? Obviously it won't move down. Wing lift comes from the low pressure at the top of the wing and the high pressure at the bottom of the wing. The downward movement of airflow is only the result of high and low pressure. Why is the top of the wing low pressure? Because the airflow tends to leave along the normal direction of the wing. Why is the bottom of the wing high? Because the airflow tends to approach along the normal direction of the wing.

A: The Newtonian explanation of flight based on the mass flow rate.
In stable cruise flight, wings with a positive angle of attack (AOA) fly through a mass of air each second (m/dt), and accelerate this air to a velocity (dv) downwards. This action crates a downward force (i.e. Force = ma = m/dt x dv). The reaction generates an equal and opposite upward force that provides lift. Lift is the vertical component of the upward force. Simply put, when the air goes down and the airplane goes up.

A: Initially air moving over the top of the wing but not touching the wing, because it is sticky (viscous), pulls air between itself and the wing top away creating a low pressure zone on top of the wing. The slope on top of the wing makes it possible to create this low pressure zone.
When air strikes the front of the wing it is compressed and then  expands into the low pressure zone with increased speed but lower pressure than the pressure in the atmosphere.
On the bottom of the wing most of the lift arises because of the angle a wing it tilted at ( angle of attack ) .This angle causes downwards deflection of air and because of Newtons Law (action reaction) the wing is pushed upwards.
A: The questioner continues objections because of other forms of flight that he points out.  If we define flight only as a body creating lift using some way of moving clean air over an airfoil, then all the airfoil discussions are totally correct, and his examples are not relevant.  If we loosen our definition of flight as getting a body off the ground for a sustain period beyond the effect of any initial ground-based propulsion, we still have balloons, rockets, and, to the point, many light aircraft with a thrust-to-weight ratio > 1, thereby allowing them to fly the aircraft stalled.  The Harrier and the F-22 are prime examples, and the Osprey can be thrown in for a discussion of why helicopters fly.
In truth, all heavier-than-air flight is a combination of at least these two simple dynamics of airfoil lift and thrust energy surplus (that reserve available after satisfying forward motion for lift).  And, of course, the whole calculus regarding wing lift gradients changes beyond the speed of sound and then at hypersonic speeds.
It is important to remember that a forward velocity is necessary for airfoil flight.  That means, without some form of internal thrust, heavier-than-air airfoil flight is only a prolonged fall through the air.  With any internal source of propulsion to sustain flight, we also give the pilot a way to create an energy surplus for maneuvering, increasing speed, or gaining altitude. Ask a pilot how he flies: "Angle-of-Attack, Air Speed, Altitude (repeat)". The airfoil is only a component.  
