A non-orientable surface is any surface that contains a Möbius band, or, strictly speaking, a subset that is homeomorphic to the Möbius band. On a non-orientable surface, there's no way to consistently define the notions of 'right' and 'left' and anything that is slid around a non-orientable surface will come back to its starting point as a mirror image.
Non-orientable surfaces form two classes, those based on the real projective plane, which have odd Euler characteristic, and those based on the Klein bottle, which have even Euler characteristic. All surfaces with an odd Euler characteristic are non-orientable. All non-orientable surfaces can be obtained by starting with the projective plane or the Klein bottle and adding handles to them – the number of handles being known as the genus.
A non-orientable surface can't be embedded (i.e. mapped one-to-one) in three-space but can be immersed (mapped locally but not globally) there.
Compare with orientable surface.