As compared to classes of graphs embeddable in other surfaces.
Some ways in which they're exceptional:
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Mac Lane's and Whitney's criteria are algebraic characterizations of planar graphs. (Well, mostly algebraic in the former case.) Before writing this question, I didn't know whether generalizations to graphs embedded in other surfaces existed, but some lucky Google-fu turned up some references — in particular there seems to be a generalization of Whitney's criterion due to Jack Edmonds for general surfaces, although frustratingly I can't find the paper, and the main reference I found implies that there might be a small problem on the Klein bottle. Anyone know if Edmonds' result is as easy to prove as Whitney's?
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Kuratowski's classic characterization of planar graphs by forbidden minors. Of course this does generalize to other surfaces, but this result is both incredibly deep and difficult (as opposed to the proof of Kuratowski, which is by no means trivial but is obtainable by a sufficiently dedicated undergraduate — actually my working it as an exercise is largely what motivated the question) and is in some sense "essentially combinatorial" in that it applies to a wider class of families that aren't inherently topologically defined.
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In the other direction of difficulty, the four-color theorem. It's apparently not difficult to show (except for the plane) that what turns out to be the tight upper bound on the chromatic number of a graph embeddable on a surface (other than, for whatever reason, the Klein bottle) is, in fact, an upper bound — the problem is showing tightness! Whereas it's pretty much trivial to show that $K_4$ is planar (to be fair, though, tightness is easy to check for surfaces of small genus — the problem's in the general case), but the four-color theorem requires inhuman amounts of calculation and very different, essentially ad-hoc methods.
I realize that the sphere has genus 0, which makes it unique, and has trivial fundamental group, which ditto, but while I imagine this information is related to the exceptionalness of the plane/sphere, it's not really all that satisfying as an answer. So, why is it that methods that work everywhere else tend to fail on the sphere?
Related questions: reasons-for-the-importance-of-planarity-and-colorability
Best Answer
(I think that the question of why planar graphs are exceptional is important. It can be asked not only in the context of graphs embeddable on other surfaces. Let me edit and elaborate, also borrowing from the remarks.)
Duality: Perhaps duality is the crucial property of planar graphs. There is a theorem asserting that the dual of a graphic matroid M is a graphic matroid if and only if M is the matroid of a planar graph. In this case, the dual of M is the matroid of the dual graph of G. (See this wikipedia article). This means that the circuits of a planar graph are in one to one correspondence with cuts of the dual graph.
One important manifestation of the uniqueness of planar graphs (which I believe is related to duality) is Kasteleyn's formula for the number of perfect matchings and the connection with counting trees.
Robust geometric descriptions: Another conceptual difference is that (3-connected or maximal) planar graphs are graphs of convex 3-dimensional polytopes and thus have extra geometric properties that graphs on surfaces do not share.
The geometric definition of planar graphs (unlike various generalizations) is very robust. A graph is planar if it can be drawn in the plane such that the edges do not intersect in their interiors and are represented by Jordan curves; The class of planar graphs is also what we get if we replace "Jordan curves" by "line intervals," or if we replace "no intersection" by "even number of crossings". The Koebe-Andreev-Thurston theorem allows to represent every planar graph by the "touching graph" of nonoverlapping circles. Both (related) representations via convex polytopes and by circle packings, can respect the group of automorphisms of the graph and its dual.
Simple inductive constructions. Another exceptional property of the class of planar graphs is that planar graphs can be constructed by simple inductive constructions. (In this respect they are similar to the class of trees, although the inductive constructions are not so simple as for trees.) This fails for most generalizations of planar graphs.
A related important property of planar graphs, maps, and triangulations (with labeled vertices) is that they can be enumerated very nicely. This is Tutte theory. (It has deep extensions to surfaces.)
It is often the case that results about planar graphs extend to other classes. As I mentioned, Tutte theory extends to triangulations of other surfaces. Another example is the fundamental Lipton-Tarjan separator theorem, which extends to all graphs with a forbidden minor.
The study of planar graphs have led to important graph theoretic concepts Another reason (of a different nature) why planar graphs are exceptional is that several important graph-theoretic concepts were disvovered by looking at planar graphs (or special planar graphs.) The notion of vertex coloring of graphs came (to the best of my knowledge) from the four color conjecture about planar graphs. Similarly, Hamiltonian paths and cycles were first studied for planar graphs.
Graphs on surfaces and other notions generalizing planarity. Considering the class of all graphs that can be embedded in a given surface is a natural and important extension of planarity. But, in fact, for various questions, graphs embeddable on surfaces may not be the right generalization of planar graphs.
David Eppstein mentioned another generalization via the colin de Verdier invariant. This describes a hiearachy of graphs where the next class after planar graphs are "linklessly embeddable graphs". Those are graphs that can be embedded in space without having two disjoint cyles geometrically link. As it turned out this is also a very robust notion and it leads to a beautiful class of graphs. (They all have at most 4v-10 edges where v is the number of vertices; The known case of Hadwiger's conjecture for graphs not having K_6 minor implies that they are all 5 colorable.) Further classes in this hierarchy are still very mysterious. Other extensions of planarity are: 3) (not literally) Graphs not having K_r as a minor; 4;5) (Both very problematic) As Joe mentioned, graphs of d-polytopes, and also graphs obtained from sphere packings in higher dimensions; 6) (not graphs) r-dimensional simplicial complexes that cannot be embedded in twice the dimension, 7) [A notion that I promoted over the years:] graphs (and skeleta) of d-polytope with vanishing second (toric) g-number, and many more.
Forbidden minors and coloring. As for the second and third items in the question. About coloring I am not sure if we should consider 4-coloring planar graphs and coloring graphs on other surfaces as very related phenomena. Regarding forbidden minors. Kuratowski's theorem on surfaces is a special case (and also an important step of the proof) of a much more general result (Wagner's conjecture proved by Robertson and Seymour) about any minor-closed class of graphs. This result can be regarded as extending Kuratowski theorem and also (and perhaps more importantly) extending Kruskal and Nash-Williams theorem on trees. Indeed Kuratoski's theorem is related nicely to the more general picture of topological obstruction to embeddibility. If you would like to propose a different (perhaps topological) understanding of the extension of Kuratowski's theorem for surfaces, then maybe you should start by the well-quasi ordering theorem for trees.