A space (such as a topological space) is second-countable if, in a certain sense, there is only a countable amount of information globally in its topology. (Change ‘globally’ to ‘locally’ to get a first-countable space.)
The weight of a space is the minimum of the cardinalities of the possible bases . We are implicitly using the axiom of choice here, to suppose that this set of cardinalities (which really is a small set because bounded above by the number of open subspaces, and inhabited by this number as well) has a minimum. But without Choice, we can still consider this collection of cardinalities.
Then a second-countable space is simply one with a countable weight.
A compact metric space is second-countable. A separable metric space, e.g., a Polish space, is second-countable. (It is not true that separable first-countable spaces are second-countable; a counterexample is the real line equipped with the half-open or lower limit topology that has as basis the collection of half-open intervals .)
For second-countable T_3 spaces , if is locally compact, then the exponential is second-countable. Cf. Urysohn metrization theorem and Polish space. I (Todd Trimble) am uncertain to what extent the assumption can be removed.