Completely regular spaces include all metrizable spaces, topological vector spaces, and topological groups in general. In fact, they are exactly the uniformizable spaces. Complete regularity is hereditary, ie. a subspace of a completely regular space is also completely regular, and it's preserved by arbitrary products. The completely regular spaces are in fact a reflective subcategory of $\mathrm{Top}$, as can be seen from another characterization: they are exactly the spaces for which the real functions determine the topology, ie. the topology is the initial topology for the set of real continuous functions. Finally, every subspace of a compact Hausdorff space is completely regular, and conversely, every ($T_0$) completely regular space embeds universally into a compact Hausdorff space via the Stone-Čech compactification.
There are of course many natural examples of topologies that aren't completely regular. The two that I know of are Zariski topology, which is $T_1$, but not Hausdorff, and Alexandrov topology, which is a natural topology on a poset that can't even be $T_1$ in an interesting way.
What I haven't seen before are examples of Hausdorff topologies that aren't completely regular, and weren't constructed specifically for the purpose of being a counterexample. Considering how widely topology is applied (and how little I know of it) I'm assuming there are some, and I'd be interested in hearing how often they appear.
The strengthening of this question to normal spaces seems to have elementary and satisfactory answers: the topology of pointwise convergence on $\mathbb R$ isn't normal (witnessing the fact that normal spaces aren't closed under products).