Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical.
In many branches of mathematics, it seems to me that a good counterexample can be worth just as much as a good theorem or lemma. The only branch where I think this is explicitly recognized in the literature is topology, where for example Munkres is careful to point out and discuss his favorite counterexamples in his book, and Counterexamples in Topology is quite famous. The art of coming up with counterexamples, especially minimal counterexamples, is in my mind an important one to cultivate, and perhaps it is not emphasized enough these days.
So: what are your favorite examples of counterexamples that really illuminate something about some aspect of a subject?
Bonus points if the counterexample is minimal in some sense, bonus points if you can make this sense rigorous, and extra bonus points if the counterexample was important enough to impact yours or someone else's research, especially if it was simple enough to present in an undergraduate textbook.
As usual, please limit yourself to one counterexample per answer.
Best Answer
The matrix $\left(\begin{smallmatrix}0 & 1\\ 0 & 0\end{smallmatrix}\right)$ has the following wonderful properties. (Feel free to add or edit; I can't remember all the reason I loathed it when I was learning linear algebra. It's funny how unexciting they all now seem, but it's a counterexample for almost every wrong linear algebra proof I tried to give.)
Only zeroes as eigenvalues, but non-zero minimal polynomial (in particular, the minimal polynomial has bigger degree than the number of eigenvalues). Probably my favorite way to state this fact: the minimal polynomial is not irreducible or square-free. The same thing in a fancier language: the Jordan canonical form is not diagonal.
Not diagonalizable, even over an algebraically closed field.
Not divisible over $\mathbb C$. There are no matrices $M$ and integers $n\ge2$ so that $M^n = \left(\begin{smallmatrix}0 & 1\\\ 0 & 0\end{smallmatrix}\right).$ All diagonalizable and most non-diagonalizable complex matrices have $n$th roots.
(This is because, if there was a square root, it'd have minimal polynomial x4, but since it's a two-by-two matrix, Cayley-Hamilton implies that the characteristic polynomial has degree 2).
The matrix is nilpotent but not zero.
It's one of the best examples when you need to remember why matrix multiplication is not commutative.
Thinking of k2 as a k[x]-module where x acts as this matrix should give wonderful (counter)-examples of modules for all the same reasons.
Also, $\left(\begin{smallmatrix}1 & 1\\ 0 & 1\end{smallmatrix}\right)$ is an example of an invertible matrix with the first three properties above. Its action on k2 is in some sense the simplest example of a representation of a group ($\mathbb{Z}$) which is indecomposable but not irreducible.