For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable.
This states that every Hausdorff second-countable regular space is metrizable.
(Historical note: The form of the theorem shown here was in fact proved by Tikhonov in 1926.
What Urysohn had shown, in a paper published posthumously in 1925, was that every second-countable normal Hausdorff space is metrizable.)
The converse does not hold: there exist metric spaces that are not second countable, for example, an uncountable set endowed with the discrete metric.
For example, a compact Hausdorff space is metrizable if and only if it is second-countable.
Urysohn's Theorem can be restated as: A topological space is separable and metrizable if and only if it is regular, Hausdorff and second-countable.
The Nagata–Smirnov metrization theorem extends this to the non-separable case.
It states that a topological space is metrizable if and only if it is regular, Hausdorff and has a σ-locally finite base.
that is, the countably infinite product of the unit interval (with its natural subspace topology from the reals) with itself, endowed with the product topology.
endowed with the strong operator topology is metrizable (see Proposition II.1 in [4]).
Examples of non-metrizable spaces Non-normal spaces cannot be metrizable; important examples include The real line with the lower limit topology is not metrizable.