Lots of good answers. I figured I'd throw in a list of non-examples, since these are pretty handy as well. (These are all standard non-examples, nothing fancy.)

A non-Noetherian ring with only one prime ideal: (k[x1, x2, x3, ...]/(xi xj : 1 <= i,j), (x1,x2,...)).

A non-Cohen-Macaulay ring: k[x, y]/(x2, xy).

A category that doesn't have products: the category of fields with field homomorphisms.

A ring which isn't flat over another ring: A = k[x2, x3] and B = k[x].

Two non-zero rings whose tensor product is zero: Z2 and Z3