Surface Classification

Every compact connected surface is homeomorphic to one of the following:
  1. a sphere
  2. a connected sum of n tori (e.g., simple n-holed donut)
  3. a connected sum of n projective planes for non-orientable case
The Euler characteristics (X)
(surface integral of Gaussian curvature)
of these surfaces is given by:
  1. a sphere has Euler characteristic 2
  2. n_tori has Euler characteristic 2-2n
  3. m_projective planes has Euler characteristic 2-m

Thus orientable surfaces can only have even Euler characteristic
whereas non-orientable surfaces can have Euler characteristic equal to any number less than or equal to 1.

E.g., the Euler characteristic of the Klein bottle is 0.

The genus of a surface (max number of independent closed cuts that leave the surface in one piece)
is defined as follows (where X is the Euler characteristic):

So the Klein bottle is non-orientable of genus 2.

This info can be found in an easy-to-read undergraduate text "Topology of Surfaces" by L Kinsey.