nLab
conformal group

Intuition

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.

Description in euclidean space

In euclidean $n$-space for $n>2$ a general conformal transformation is some composition of a translation, dilation, rotation and possibly an inversion with respect to a $n-1$-sphere. For $n=2$, 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 $z$ and $\overline{z}$ as independent coordinates in the complexification ${ℂ}^2$ and restricts to the real part ${ℝ}^2\cong ℂ$). This is important for CFT in 2d.