TITLE: A geometric reference for (affine) Gorenstein varieties and singularities 
QUESTION [15 upvotes]: I would like to ask for a reference to some text that explains in relatively down to earth (if possible geometric) terms (for dummies) what is a Gorenstein singularity and Gorenstein variety (for a person whose knowlage of commutative algebra is minor). What standard things one could try to do to check if a given scheme is Gorenstein?
If the field of definition is $\mathbb C$ - it is even better. I know one source - Eisenbud "commutative algebra" but find it a bit hard.

REPLY [10 votes]: Sorry, tough luck, but most first (and second) algebraic geometry courses don't even touch Cohen-Macaulay rings, let alone Gorenstein. Look, for example, at Definition 4.2 here. So it is unlikely you can find such reference.
Here is an explanation why you need both Cohen-Macaulayness and the fact that the canonical divisor is Cartier, as mentioned in Karl's answer. The trouble is that there are UFDs (so all divisors are Cartier) which are not  Cohen-Macaulay (for instance, the invariant ring of  $\mathbb Z_4$ acting by cyclically permuting the variables on the polynomial ring in four variables over a field of char 2). Such examples are not very well-known, I remember Sándor Kovács pointed out in a recent comment that most people don't even realize that it could be an issue.
But without Cohen-Macaulayness, the canonical sheaf would not be truly dualizing (see the comments here). This is perhaps where the real power of the property lies.
Finally, I would like to recommend the survey on Gorenstein rings by Craig Huneke: Hyman Bass and Ubiquity: Gorenstein Rings. You can pick up a lot about them from there, including the very interesting history. To quote from the Introduction:

As we shall see, they could  perhaps more justiﬁably be called Bass rings, or Grothendieck rings, or Rosenlicht  rings, or Serre rings.

Enjoy!