Topological spaces vs. metric spaces
Ordinal spaces occur naturally, and any uncountable ordinal space is not metrizable.
You can also talk about Moore spaces, Stone-Cech compactification of $\Bbb N$ (which is compact but has is too big to be metrizable), there are Zariski topologies which are often non-metrizable, and there are plenty of Cantor cubes which are naturally occurring in set theory.
Many important quotient spaces are not metric, nor even Hausdorff. This happens quite commonly for orbit spaces of group actions, which are very important in geometric group theory, in dynamical systems, etc.