Möbius strip => A 2 Dimensional strip twisted 180° and joined at the ends. If you move along the surface, you return to your starting point on the "other side" — thus there's only one side.
Klein vase => An extension of the Möbius strip into a closed 3 Dimensional shape (or actually a 4D shape), which loops back into itself so that the inside and outside are the same surface and thus there is only one side.
