This is certainly related to "What are your favorite instructional counterexamples?", but I thought I would ask a more focused question. We've all seen Counterexamples in analysis and Counterexamples in topology, so I think it's time for: Counterexamples in algebra.
Now, algebra is quite broad, and I'm new at this, so if I need to narrow this then I will- just let me know. At the moment I'm looking for counterexamples in all areas of algebra: finite groups, representation theory, homological algebra, Galois theory, Lie groups and Lie algebras, etc. This might be too much, so a moderator can change that.
These counterexamples can illuminate a definition (e.g. a projective module that is not free), illustrate the importance of a condition in a theorem (e.g. non-locally compact group that does not admit a Haar measure), or provide a useful counterexample for a variety of possible conjectures (I don't have an algebraic example, but something analogous to the Cantor set in analysis). I look forward to your responses!

You can also add your counter-examples to this nLab page: http://ncatlab.org/nlab/show/counterexamples+in+algebra
(the link to that page is currently "below the fold" in the comment list so I (Andrew Stacey) have added it to the main question)

In the category of rings, epimorphisms do not have to be surjective:
$\mathbb{Z}\hookrightarrow \mathbb{Q}$.