A Klein bottle is a three-dimensional Mobius loop. Brief explanation:
A Mobius loop is made by taking a two-dimensional strip, giving it a half-twist, and joining the ends. It only has one edge, and the left is the same as the right. The half-twist requires the use of the third dimension.
A Klein bottle is made by taking a three-dimensional cylinder, giving it a half-twist, and joining the ends. It only has one face, and the inside is the same as the outside. The half-twist requires the use of the fourth dimension.
Obviously, since we can't use the fourth dimension to make a Klein bottle, the bottle intersects itself. (A Mobius strip would also intersect itself if you only had two dimensions to make it in.)
CP can probably wax more poetic on the coolness of the Klein bottle.
