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: <code>(k[x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>, ...]/(x<sub>i</sub> x<sub>j</sub> : 1 &lt;= i,j), (x<sub>1</sub>,x<sub>2</sub>,...))</code>.

A non-Cohen-Macaulay ring: <code>k[x, y]/(x<sup>2</sup>, xy)</code>.

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

A ring which isn't flat over another ring: <code>A = k[x<sup>2</sup>, x<sup>3</sup>]</code> and <code>B = k[x]</code>.

Two non-zero rings whose tensor product is zero: <code>Z<sub>2</sub></code> and <code>Z<sub>3</sub></code>