The one-dimensional case is well known: you have the circle, the line $\mathbb{R}$, the long line $L$ and the long ray $R$. The proof is not that easy to find in the literature since non-metrizable manifolds are (in my opinion) underestimated. It is described in a previous answer, so let me give a few pointers for the two-dimensional case. Before that, let me stress that there is another hypothesis in the usual definition of manifolds that is often overlooked: being Hausdorff. Non-Hausdorff manifolds are interesting too because they appear naturally, e.g. space of leaves of a foliation. I do not know if anyone has studied non-Hausdorff non-metrizable manifolds. That would really be looking for trouble.

Back to the point. There is very, very little hope to classify non-metrizable surfaces: there are many of them, and of various kind.

First one think about easy examples: product of two one-dimensional non-metrizable manifolds; that makes 4 examples.

*But there are more*: take the first octant of the product of two long lines (i.e. $\{(x,y)\in R\times R | y \le x \}$). Or glue a bunch of octants together along their edges. This already makes you a (small) bunch of examples. At that point, I should stress that in the product of two long lines, the diagonal is very different from the coordinate axes. Each axis $A$ is homotopic to a copy of $L$ that is disjoint from $A$, while any embedding of $L$ which is homotopic to the diagonal must meet it on an unbounded set. This comes from the homotopy theory of $L$ and $R$: e.g. there are two homotopy classes of maps $R\to R$: the ones homotopic to a constant, which are exactly the bounded maps, and the ones homotopic to identity, which have an unbounded set of fixed points.

*But there are more*: one can produce many different "long pipes", which are obtained as increasing unions of annuli indexed by $\omega_1$, where at each non-limit ordinal the inclusion is as trivial as one might want, but at each limit ordinal the lower annuli can accumulate to a circle with a segment attached, or to something worse. By choosing the shape of such singularities, and at which limit ordinal they do appear gives you a very large range of long pipes.

The good news is that classifying long pipes is sufficient to get a classification of $\omega$-bounded surfaces (i.e. surfaces in which any countable sequence has an accumulation point; e.g. the long line is $\omega$-bounded but the long ray is not). This is thank to the beautiful "bagpipe theorem" of Nyikos (The theory of non-metrizable manifolds, in K. Kunen and J. Vaughan, eds, “Handbook of Set-Theoretic Topology” (Elsevier, 1984), 633–684) which says that any $\omega$-bounded surface is obtained by gluing finitely many long pipes (the pipes, obviously) to a compact surface with some disks removed (the bag). The bad news is that even a classification of long pipes seems out of reach. If I remember well, it is an open question whether every long pipe contains an embedded long line.

The worse news is: *but there are more*. $\omega$-bounded surfaces are a very particular kind of surfaces. A non-metrizable surface which is very different from everything above is the Prüfer manifold. Basically, you glue a bunch (i.e., one for each real number) of planes to a half plane in a way that maps half infinite strips to cones, so that the different planes do not interfere two much one with the other. This is a huge, weird space.

But I guess that *there are more* (if I remember well, it has been proved that there are $2^{\aleph_1}$ pairwise non-homeomorphic non-metrizable surfaces, but I do not know in which axiom system it holds).