I read about Du Val singularities on surface are classified by equations of ADE type. For example, $x^2+y^2+z^{n+1}=0$ for A type. As not every surface can have a neighbourhood embedded in $\mathbb{A}^3$. Why do we use analytic coordinates to characterize singularity? 



Why do we use analytic coordinates to characterize singularity?
The point is that, *locally*, any Du Val singularity (that is, any isolated surface singularity that arises by contracting an $A$-$D$-$E$ curve) can be realized as a double cover of a nonsingular surface. This means that there exist local analytic coordinates such that the germ of singularity has the form $$x^2=f(y, \, z),$$
that is the corresponding embedding dimension is $3$. In other words, even if the surface is not globally embeddable in $\mathbb{A}^3$, an analytic neighbourhood of the singularity always is. 


A good reference is [Barth-Peters-Van De Ven, *Compact Complex Surfaces*, Springer 1984], see in particular Lemma 3.8 of Chapter III.


