Theorem
A topological space $X$ is metrisable if and only if it is regular, Hausdorff and has a countably locally finite base.
A metrization theorem is a result that gives sufficient conditions, and sometimes necessary and sufficient conditions, for a topological space to be metrisable, i.e. its topology is induced by a metric.
An optimal metrization theorem, giving a necessary and sufficient, condition is Nagata-Smirnov metrization theorem:
A topological space $X$ is metrisable if and only if it is regular, Hausdorff and has a countably locally finite base.
A variation of this, directly implied by the fact that metrisable spaces have countably locally discrete bases, is the Bing metrization theorem:
A topological space $X$ is metrisable if and only if it is regular, Hausdorff and has a countably locally discrete base.
A historical predecessor and direct implication of these theorems is the Urysohn metrization theorem:
Every second-countable regular Hausdorff space $X$ is metrizable
Another variation is the Moore metrization theorem.
A refined and more complicated question is whether or not the topology of a given space is induced by a complete metric. In this case the space is called topologically complete. A sufficient and neceissary criterion can be given in terms of cotopology.
If one weakens the concept of metric space sufficiently to the notion of approach space, then every topological space is metrisable in this weaker sense.
James Munkres, Topology (2nd edition), Prentice-Hall, 2000.
Ryszard Engelking, General Topology, Heldermann Verlag Berlin, 1989.
Wikipedia, Metrization theorem. (link)