It is also related to the sum of indices of a vector field with isolated zeros, by the Poincare-Hopf theorem.  
It is a topological invariant which is related to the curvature of a surface, a local geometric quantity, through the Gauss-Bonnet theorem.  It is computed as an alternating sum of Betti numbers.  It is also related to the sum of indices of a vector field with isolated zeros, by the Poincare-Hopf theorem.  For a surface of genus g, it is equal to two minus two g.  FTP, what is this characteristic of a surface, which for polyhedra is equal to two and given by vertices minus edges plus faces?
It is a topological invariant which is related to the curvature of a surface, a local geometric quantity, through the Gauss-Bonnet theorem.  
Euler characteristic