The Klein bottle is a closed non-orientable (i.e. one-sided) surface. Consider, in the plane, the square of all points with Cartesian coordinates (x,y) such that −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1. Identifying each point (x,1) on the top edge with the point (x,−1) opposite, for all x, forms a cylinder with two bounding circles. Identifying each point (1, y) with (−1,−y) glues these circles in reverse orientations. The Klein bottle is constructed by doing these two operations simultaneously. This is not practically possible in three dimensions without the Klein bottle intersecting itself but can be achieved in four dimensions. Note that the shaded area in the figure is a Möbius strip.

A Klein bottle made from a square

Self-intersecting Klein bottle in 3D

http://www.kleinbottle.com/whats_a_klein_bottle.htm A fuller description and images of Klein bottles.