The McKay correspondence mentioned in Hailong's and Mike's answers extends to maximal Cohen-Macaulay modules over the invariant rings $R=k[x,y]^G$, where $G$ is a finite subgroup of $GL(2,k)$ (with $|G|$ invertible in $k$). In particular (Herzog) all such subrings have only finitely many non-isomorphic indecomposable MCM modules, that is, they have finite CM type. The converse is true in characteristic zero -- that a two-dimensional complete normal domain over $\mathbb{C}$ of finite CM type is a ring of invariants -- by a result of Auslander.

The case $G \subset SL(2,k)$ corresponds to $R$ being Gorenstein, and in particular a hypersurface -- namely the ADE hypersurfaces listed in Mike's answer. The correspondence between the irreducible representations of $G$ and the components of the exceptional fiber extends to include the irreducible MCM modules, and the McKay quiver (aka Dynkin diagram) is the same as the stable Auslander-Reiten quiver. The modules are directly connected to the irreps by Auslander, and directly connected to the components of the fiber by Gonzales-Sprinberg--Verdier and Artin--Verdier, extended to the non-Gorenstein case by Esnault and Wunram.

Most of this is in the current draft of my notes with Roger Wiegand on MCM modules, chapters 4 and 5. (Ignore the geometry in Chapter 5 -- it's riddled with errors and I'm currently rewriting it. Or you're welcome to point out errors I might not have noticed yet.) The question of what happens to the Auslander-Reiten-McKay correspondence for $G \not\subset SL(2)$ is addressed in some recent papers of Iyama and Wemyss. (You only get some of the indecomposable MCMs, the so-called special ones.)

The McKay correspondence mentioned in Hailong's and Mike's answers extends to maximal Cohen-Macaulay modules over the invariant rings $R=k[x,y]^G$, where $G$ is a finite subgroup of $GL(2,k)$ (with $|G|$ invertible in $k$). In particular (Herzog) all such subrings have only finitely many non-isomorphic indecomposable MCM modules, that is, they have finite CM type. The converse is true in characteristic zero -- that a two-dimensional complete normal domain over $\mathbb{C}$ of finite CM type is a ring of invariants -- by a result of Auslander.

The case $G \subset SL(2,k)$ corresponds to $R$ being Gorenstein, and in particular a hypersurface -- namely the ADE hypersurfaces listed in Mike's answer. The correspondence between the irreducible representations of $G$ and the components of the exceptional fiber extends to include the irreducible MCM modules, and the McKay quiver (aka Dynkin diagram) is the same as the stable Auslander-Reiten quiver. The modules are directly connected to the irreps by Auslander, and directly connected to the components of the fiber by Gonzales-Sprinberg--Verdier and Artin--Verdier, extended to the non-Gorenstein case by Esnault and Wunram.

Most of this is in the current draft of my notes with Roger Wiegand on MCM modules, chapters 4, 5, and 6. (Ignore the geometry in Chapter 5 -- it's riddled with errors and I'm currently rewriting it. Or you're welcome to point out errors I might not have noticed yet.) The question of what happens to the Auslander-Reiten-McKay correspondence for $G \not\subset SL(2)$ is addressed in some recent papers of Iyama and Wemyss. (You only get some of the indecomposable MCMs, the so-called special ones.)