Examples of non-metrizable spaces
From my perspective as an analyst, non-metrizable spaces usually arise for one of the following reasons:
Separation axiom failure: the space is not, e.g., normal. This mostly happens when the space is not even Hausdorff (spaces that are Hausdorff but not normal are usually too exotic to arise much). Often this is for simple reasons; for example, the topology is defined by a pseudometric such as a seminorm. In this case we usually mod out by zero-distance pairs and try again.
The space is too big: examples like the uncountable ordinal and the long line fall under this heading. But they are still locally metrizable, so we usually allow them if we're trying to prove local theorems, but forbid them for global statements.
The topology is too weak, and usually not even first countable, so sequences are not enough to define the topology. Most of these examples are based on a product or pointwise-convergence topology: the weak-* topology on the dual $X^{\ast}$ of a Banach space $X$, the spectrum of a $C^{\ast}$-algebra, Stone-Čech compactifications, etc. (And actually, in the first case, it is often enough just to look at the unit ball of $X^*$, which is metrizable if $X$ is separable, which in applications it usually is.) As compensation, some corollary of Tychonoff's theorem gives us some compactness, which is probably the only reason we've agreed to put up with such an annoyingly weak topology in the first place.
Someone is abusing the language of topology, e.g. Fürstenberg's "topological" proof of the infinitude of the primes.
I've wandered into an algebraic geometry seminar by mistake. Grouchier analysts may consider this to fall under the previous heading. ;-)
Every finite topological space which is not discrete is not Hausdorff and hence not metrizable. Yet, there are lots of finite topological spaces: they are in one-to-one correspondence with finite preordered sets.
I'm not an expert, but I believe the space of Distributions (in any number of variables), (a.k.a. generalised functions, including the Dirac delta, its derivatives, etc.) as used in PDE theory, is a topological vector space, but non-metrisable; even though sequences are sufficient to do everything.
However, the subspace of tempered (Schwartz) distributions, as used in Fourier Analysis, is metrisable; it is a Fréchet space.