The equation χ = 2-2g tells us the genus g of the surface (assuming closed and orientable) if we know the Euler characteristic χ, which is usually defined as χ = V - E + F, where V, E, and F are the number of vertices, edges, and faces, respectively of a polygonal representation of the surface.

You can also relate χ to the curvature of the surface via the Gauss-Bonnet theorem, but that's starting to get fairly advanced.