According to Wikipedia lift in an aircraft is due to an area of low pressure formed above the wings of an aircraft due to the fast moving air there. So why exactly is an area of low pressure created due to fast moving air?
It's all about conservation of momentum, $F=ma$. Fluid can only change velocity by experiencing a force, and the only force it can feel is a pressure difference. So if there's a velocity difference, there's a pressure difference, and vice-versa (neglecting other things, like gravity). Read this beautiful exposition.
Because when a stream of moving air attached to a curved wing is made to change direction, it sucks. The air wants to move in a straight line, but the surface underneath it is curving away from the flow. So it tends to create a partial vacuum.
If wikipedia says that, it's wrong. There is no physical requirement for the air to move faster or slower on either side of the wing. The best analogy I've found so far is to think of two layers of cotton batting, or two blankets, of infinite length. The aircraft wing separates them, but there is no need for the blankets to "line up" after the wing has passed.
If you read any half-decent text on aerodynamics, you will find that lift is almost 100% due to the angle of attack. The air below the wing is forced downwards, so momentum conservation forces the wing upwards. Aircraft can and have flown just find upside down.
The shape of the wing, with camber and trailing edge taper, etc., is almost entirely designed to minimize drag, eliminate stall vortices, and all sorts of other messy mathematical problems in turbulent flow. Lift is absolutely NOT due to reduced air density on top of the wing.
First, we need to talk define some terms in a way that is useful to this particular discussion. These are not the most common general definitions.
Temperature is a description of the amount of kinetic energy a substance has. It's the average speed (not velocity) of the particles of air. In "still" air, these particles are moving in totally random directions. Some of them are going really fast, some of them are going slow, some of them are around the average. Average speed is some number proportional to temperature, average velocity is 0 (because the particles are moving in random directions).
Pressure is a force exerted by a substance's particles on a surface. In this case, it's a result of air particles that happen to traveling towards a surface striking it and bouncing off.
Consider a cube of air. For simplicity, let's pretend all particles have exactly the average speed, "T".
When air is still, particles are moving in random directions, and each wall gets hit by particles with about the same frequency and same speed. Each wall experiences the same pressure.
Now let's make the air all move to the right, but keep the temperature the same. Now the average speed of particles is the same, but the direction they're traveling is no longer completely random. They're all moving a little to the right in addition to the direction they were going. Particles that were moving directly left are moving left a little slower. Particles moving to the right are moving to the right a little faster. Particles that were moving down are now moving down and to the right, at an angle. It's that angle that's really important here. A particle that would have hit the bottom wall with "T" speed is now hitting it with speed=sin^-1(speed to the right/T)*T, which is less than T.
The left, side, top, and bottom walls are getting hit with less force (less perpendicular particle speed), while the right wall is getting hit harder. In a closed box, this can't last, but in the case of the airplane wing, most of the walls are open.