Non-orientability of the Möbius strip
The intuitive idea of orientation of a surface next to a point is quite simple, although its formal mathematical translation is difficult at an elementary level, at least if one is using a more general concept of surface (called topological). Essentially, next to each point in the plane we can consider two orientations, which can be represented by small arcs with directions marked by small arrows: and the opposite one . But there is a subtle care whish must be taken: the orientation next to a point is determined by one of the two possible directions in a small arc next to that point. That arc and that direction are on the surface and the orientation of the surface, determined next to that point by the choice of such an arc, does not change if we "look" at the surface and that arc from the one side or the other of the surface (see the conventions on paper representations of surfaces).
The orientation of the surface determined by the curve in green does not depend on the side from which the surface is seen.
If we leave a small fixed arc and move the other in any way in the plane, we cannot return with the orientation changed; in the plane there is no path which changes the orientation: mathematicians express this by saying that the plane is orientable. Once an orientation is chosen (from the two possible ones) next to a point, we can "transport" it without ambiguity to a neighbourhood of any other point in the plan.
However, the same does not happen with the Möbius strip; on it there are paths which conserve orientation and paths which change it: the Moebius strip is not orientable. It is preferable to understand the context of representations of surfaces by physical models before seeing the animations corresponding to possible disorientation paths that can be seen from the table of links.