RE: Thin Airfoil Theory
Note: I'm omitting equations and explaining things in terms that are hopefully understandable, because, well, no one likes equations.
There's some good reading on this topic here:
Thin Airfoil Theory
Basically, thin airfoil theory is a way of approximating the performance of a given airfoil to get the lift and moment coefficients for both symmetric and cambered airfoil.
Assumptions:
1. Thickness is neglected, but the derivation is still considered relevant for airfoils with a thickness-to-chord ratio of 10% or less.
2. Inviscid (approximately valid for Re >> 1)
3. Irrotational (fluid lines do not curl; they are parallel to one another at any given position at any moment in time)
4. Incompressible (that rules out anything moving faster than about M=0.3)
Basically it is only applicable to 2-D airfoils, not 3-D finite wings. This in turn neglects induced drag from wingtips, as well as a handful of other tip and root-interference related factors.
Basic steps to the derivation:
1. Place a vortex sheet along the chord line of an airfoil.

2. Calculate the variation of the vortex strength per unit length along the vortex sheet, such that the camber line becomes a streamline of the flow and the Kutta condition is satisfied at the trailing edge (vorticity strength = 0 at the trailing edge).

3. Obtain the velocity distribution due to vortex strength along the length of the airfoil.
4. Find pressure as a function of velocity.
5. Calculate the lift coefficient and the moment coefficient for the airfoil.
Conclusion
Thin airfoil theory provides a simple, practical, reliable method of calculating airfoil properties, within the limitations of its assumptions. It is extremely useful for designing and analyzing airfoils. All you need is the mean camber line for the airfoil, and you can calculate all of the important properties of that airfoil. The major weakness of thin airfoil theory is its inability to calculate drag; that must come from later analysis.