How does the wings provide the lift to the aeroplane?

A wing creates lift by imparting a downward momentum to the air flowing above and below it. The rate of change of momentum is equal to Force (Newton's 2nd law), and therefore a reaction force pushes the wing up, producing lift (Newton's 3rd law).

The imparting of this downward momentum ("downwash") to the air results from an air pressure differential above/below the wing. If you know the pressure above the wing and the pressure below the wing, and the wing area, you can calculate the lift force on the wing since Force = Pressure x Area. If you don't know the pressures, you can get a rough estimate if you know what the average air velocities are above and below the wing. A wing creating lift will have higher speed air flowing over the top of it than flowing below it. By employing Bernoulli's Principle, you can calculate a pressure difference corresponding to the difference in velocity.

An airfoil shape is effective in generating lift since it helps to keep the air flowing smoothly around the wing, making the wing more effective in diverting the air downwards. Air tends to flow more smoothly around curved shapes rather than abrupt sharp edges which is why the top of an aircraft wing always is curved. Even a perfectly flat wing can create lift (such as in a toy balsa wood glider). However a flat wing isn't practical for a full sized airplane since it's not quite as effective in producing lift. The air doesn't flow smoothly past the sharp leading edge, resulting in a lot of drag and an abrupt stall.

Note that there is no requirement that the air molecules separating at the leading edge and flowing below the wing meet up with the same molecules that flow over the top. This is called the "equal transit time theory" and is a popular science myth that unfortunately has found it's way into flight manuals and even some undergraduate texts. However, aerodynamics engineers have known ever since they started doing wind tunnel testing that the air flowing over a lifting wing reaches the trailing edge sooner than the air below it. This is true even for a perfectly flat wing. This can be explained in terms of the circulation theory, which is an advanced concept.