In cartography, a conformal map projection is a map projection that preserves the angles at all but a finite number of points. Examples include the Mercator projection and the stereographic projection.
Differential geometry
In Riemannian geometry, two Riemannian metrics g and h on smooth manifold M are called conformally equivalent if for g=uh for some positive function u on M. A diffeomorphism between two Riemannian manifolds is called conformal if the pulled back metric is conformally equivalent to the original one. The conformal maps preserve angles and shapes of (infinitesimally) small figures. For example, stereographic projection of sphere onto the plane augmented with a point at infinity is a conformal map.
One can also define a conformal structure on a smooth manifold to be class of conformally equivalent Riemannian metrics.
Any conformal map from Euclidean space to it-self is a compositins of homothety and isometry.
Complex analysis
An important family of examples is coming from complex analysis, a function f : U -> C (where U is an open subset of the complex plane C) is conformal if and only if it is holomorphic or antiholomorphic (i.e conjugate to holomorphic) and its derivative is everywhere non-zero. The Riemann mapping theorem states that any open simply connected proper subset of C admits a conformal map on open unit disk in C.
A map of the extended complex plane (which is the conformally equivalent to a sphere) onto itself is conformal iff it is a Möbius transformation or its conjugate.