What exactly does a Möbius transformation do? From what I understand, a Möbius transformation is of the form f(z) = $\frac{Az+D}{Cz+B}$ where A,B,C, and D may be real or complex. What is f(z) doing to z exactly? And what are some of the significance of this?
I am also a little confused about the difference of real and complex numbers in a Möbius transformation. What are some differences if the variables are real or complex?
 A: Any Möbius transformation is a composition of translation, rotation, reflection, homothety (scaling) and inversion, see here. The first three are straightforward to picture, and the last one is an analog of reflection, only across a circle rather than a line, a picture here gives the idea. Their properties of mapping lines and circles into lines and circles, and mapping "point at infinity" into a finite point are used extensively in complex analysis.
If $A,B,C,D$ are real then real numbers $x$ will be mapped into real numbers $f(x)$ or infinity, i.e. the transformation preserves the extended real line. Moreover, the (extended) upper half plane and lower half plane, that real line separates, get mapped into themselves. The upper half-plane (with a non-standard way to measure distances) is a model of hyperbolic geometry, the Poincaré half-plane model. In this model real Möbius transformations are the isometries, i.e. they preserve the hyperbolic distances, so they are 'translations, rotations and reflections' of the hyperbolic plane. 
Complex Möbius transformations map the upper half plane into other domains that serve as different models of hyperbolic geometry, e.g. Poincaré disk model. Another group  of complex Möbius transformations preserve that disk and serve as hyperbolic isometries of it. They produce spectacular tilings, used in art by Escher, that are created by starting with a seed pattern and spreading it around using hyperbolic isometries like you would use ordinary translations, rotations and reflections in the Euclidean plane.
