In several courses of algebra, I've heard that not all PIDs are EDs, and the canonical example is $\mathbb{Z}\left[\dfrac{1+\sqrt{-19}}{2}\right]$ which I've heard over and over. Some cursory research both here and elsewhere has not turned up other examples of non-Euclidean principal ideal domains.
Where might I find resources describing them, or can anyone provide examples?
Another nice example of a non-Euclidean PID is $ K[[x,y]][1/(x^2\!+\!y^3)]\,$ over a field $\,K,\,$ i.e. adjoin the inverse of $\,x^2\!+\!y^3$ to a bivariate power series ring over a field. The proof employs a general construction method, see
D. D. Anderson. An existence theorem for non-euclidean PID’s,
Communications in Algebra, $16:6, 1221\!-\!1229, 1988$.
Also worth keeping in mind is that for number rings, a result of Weinberger (1973) (assuming GRH) implies that a UFD number ring R with infinitely many units is Euclidean. Therefore, e.g. real quadratic number rings are Euclidean $\!\iff\!$ PID $\!\iff\!$ UFD.