Several other metrization theorems follow as simple corollaries to Urysohn's theorem. For example, a compact Hausdorff space is metrizable if and only if it is second-countable.
The real line with the lower limit topology is not metrizable. The usual distance function is not a metric on this space because the topology it determines is the usual topology, not the lower limit topology. This space is Hausdorff, paracompact and first countable.
The long line is locally metrizable but not metrizable; in a sense it is "too long".