Higher dimensional analog to planar graphs?

A planar graph is nothing other than a triangulation of the $2$-sphere (via stereographic projection), at least if one allows arbitrary polygons instead of just triangles, so a natural generalization to three dimensions is triangulations of the $3$-sphere where one allows arbitrary (convex) polyhedra instead of just simplices. One can of course talk about triangulations of arbitrary manifolds.

Another generalization is along the genus axis, i.e., consider which graphs can be embedded into a 2d compact surface, such as an $n$-torus. This area is called topological graph theory.