This page provides an introduction to many topological terms and concepts.
A surface is a 2 dimensional manifold.
Non-orientability is an intrinsic property of manifolds. In non-orientable surfaces, an object within the surface can travel along a path that will lead it back to its start point, but with its right and left sides flipped.
Here is a more technical explanation: In a non-orientable manifold, there exists at least one path around the manifold such that, if we take a set of basis vectors for the manifold and move them along this path, they will arrive at their starting point with the following result. The determinant of the matrix composed of these basis vectors will have the opposite sign (positive/ negative) when it arrives back at the starting point as it did when it left.
Not all image pages will discuss in depth how non-orientability arises for a particular surface; the phenomenon is discussed, however, for the Mobius Strip and Real Projective Plane, with images to explain the phenominon.