The wikipedia article claims that the theorem "was first proved by G. C. Shephard and J. A. Todd (1954) who gave a casebycase proof. Claude Chevalley (1955) soon afterwards gave a uniform proof". I read the paper by Chevalley and it seems that he only proves the implication: "If the group is generated by pseudoreflections, then the ring of invariants is polynomial". I wonder whether there is a uniform proof of the inverse implication? Where is it written?
There are indeed many presentations (if I remember correctly Bourbaki has it) but the proof is very elegant and short so that I find it hard to refrain from giving it. Let $H$ be the normal subgroup of the finite $G\subset \mathrm{GL}_n$ generated by the pseudoreflections. By the other direction $X:=\mathbb{A}^n/H$ is again affine space and in particular is smooth. We have an action of $G/H$ on $X$ and a moment's thought reveals that it acts freely in codimension $1$ (as a point fixed by a nonidentity element would lie below a reflection hyperplane of $\mathbb{A}^n$ and the fixing element below a pseudoreflection). Hence $X \to X/(G/H)=\mathbb{A}^n/G$ is étale in codimension $1$. If $\mathbb{A}^n/G$ were smooth, purity of the branch locus would imply that the map were étale. However, that forces $G/H$ to act freely on $X$ but the image of the origin is fixed by all of $G/H$ and therefore $G=H$. 


Chevalley was interested in the action of (real) Weyl groups and so a reflection to him had determinant 1 and so was a real reflection, i.e. order 2. My understanding is that Serre had seen the paper by Shepard and Todd and so he knew that pseudoreflections were relevant. He pointed out that Chevalley's proof was valid for pseudoreflections. 


$\mathbb{C}$
, leaving aside the more delicate question of what remains true over more general fields mentioned here by Victor Protsak. This book treats ShephardTodd theory and later developments in good detail. – Jim Humphreys Jul 19 '10 at 10:41