A space (such as a topological space) is first-countable if, in a certain sense, there is only a countable amount of information locally in its topology. (Change ‘locally’ to ‘globally’ to get a second-countable space.)
The character of a space at a point 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 neighbourhoods of , and inhabited by this number as well) has a minimum. But without Choice, we can still consider this collection of cardinalities.
Then a first-countable space is simply one whose characters are all countable.
The character, tout court, of a space is the supremum of the characters of its points; then a first-countable space is simply one with a countable character.
Any second-countable space must also be first-countable.
Any metric space is first-countable.