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: