I'm simplifying here, but every introduction to flying shows us that the profile of a wing leads to lower pressure on the upper side of the wing, hence the wing and the plane attached to it will be pulled up.
All right.
How does that explain a plane flying inverted? If the explanation was right, the plane would pull itself towards earth.
Normally, an airfoil is optimized for best lift/drag (L/D) efficiency for a certain flight profile (usually a compromise). Since most of the time for most aircraft inverted flight is not an issue, you get an airfoil that is optimized for upright flight, and this is best achieved with asymmetric geometries.
However, depending on the angle of attack any airfoil can (and will) generate "negative" lift, only much less efficiently so, than for the optimized regime, resulting in increased drag.
The desired angle of attack for conventional aerodynamically controlled aircraft is maintained by the elevator. For symmetric airfoils commonly used for aerobatic planes, the performance for upright and inverted flight is quite similar. For 99% of all other airfoils inverted flight will work up to a certain point, depending on the available power, CG, the maximum lift and rudder forces availabe before stall. As a result, for some aircraft, a stable inverted flight cannot be maintained, while for others it could be (but with varying penalty to performance, stall speed etc). The aerodynamical possiblity of inverted flight is of course limited by structural and other considerations.
This relation between a wing's curvature and a pressure difference on both sides is often part of the 'equal transit time' explanation; air on the curved side has to traverse a longer distance in the same amount of time, therefore goes faster, which leads to a lower pressure. This explanation is very common and completely wrong.
In normal flight, pitching the nose up causes the aircraft to climb because the wings meet the air at a steeper angle; the lift increases. It makes sense that rotating the wings in the opposite direction decreases lift. In fact, point the nose down far enough and the wings will produce no lift at all. Beyond that, the generated lift becomes negative and the wings will start to pull the aircraft down.
During our hypothetical manoeuvre, our attitude has varied by about 10°. That's not exactly flying upside down yet, the curved side of the wings were on top the entire time. Whether or not the lift was pointing up as well, depended on the angle at which the wings meet the air, the angle of attack.
The same is true for inverted flight. If we find ourselves at an attitude where the wings are pulling us down, we raise the nose. At first, the downward lift will disappear and at higher angles of attack, start pointing up and grow larger. At sufficient airspeeds and angles of attack, we have enough lift to maintain altitude upside down.
So why do wings need to be curved at all? They don't. Flat wings also provide lift at non-zero angles of attack and are perfectly usable, but not very efficient. Properly shaped airfoils create more lift and less drag. To find out why, consult a more accurate explanation of how planes really fly.
Short answer is that explanation is wrong. How planes actually generate lift is far more complicated.
Bernoulli's Principle (airfoil shape) is only one of the lifting forces.
Equally important is deflection (Newton's law), and on propellor aircraft, accelerated airflow.
'Newtonian' lift is the upward reaction of the wing to its downward deflection of the air-stream. The most efficient way to deflect the airstream is a gradual acceleration, accomplished by a concave lower surface. The upper surface shape must prevent premature 'separation' (chaotic vacuum eddies) of the upper airflow. A symmetric wing can still deflect the air-stream, depending on its angle-of-attack; it has the same lift and drag whether right-side-up or inverted, which is not as efficient as a right-side-up normal wing, but better than an inverted normal wing.
