A conformal transformation (conformal mapping) is a transformation of a space which preserves the angles between the curves. In other words, it preserves the angels infinitesimally. Conformal group of a space which has well defined notion of angles between the curves is the group of space automorphisms which are also conformal transformations.
In euclidean -space for a general conformal transformation is some composition of a translation, dilation, rotation and possibly an inversion with respect to a -sphere. For , i.e. in a complex plane, this still holds for (the group of) global conformal transformations but one also has nontrivial local automorphisms. One has in fact infinite-dimensional family of local conformal transformations, which can be described by an arbitrary holomorphic or an antiholomorphic automorphism (in fact one writes and as independent coordinates in the complexification and restricts to the real part ). This is important for CFT in 2d.