Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.
A metric space is a generalisation of distance, and is a set for which the distance between all elements of the set are defined. Some different type of metric space include
1) Complete metric spaces (every Cauchy sequence converges)
2) Bounded metric spaces (every metric is bounded by a finite value)
3) Compact metric spaces (every sequence has a convergent subsequence)
4) Locally compact metric spaces (every point has a compact neighbourhood)
5) Separable metric spaces (it possesses a countable dense subset)