Möbius Band:

You can make one by taking a long thin strip of paper and attaching the two short ends together so that the top of one is at the bottom of the other:

A Möbius band has a single boundary curve, and only one side. It is non-orientable, which means that it contains a path for which it is impossible to define a left- and right-hand side in a consistent global way. In fact, it is the fundamental unit of non-orientability in that any non-orientable surface contains a Möbius band.

A Möbius band can not be embedded in the plane.

See also:

The real projective plane

The Klein bottle