The Klein bottle is a surface with only one face. It is non-orientable, i.e. one cannot distinguish between inside and outside.
Topologically, it can be obtained by first glueing a unit square to a tube and then identifying both ends of the tube after reversing the orientation. In order to realize the Klein bottle in three-dimensional space, it needs to self-intersect, whereas it can be embedded into four-dimensional space.
The Klein bottle was first described in 1882 by Felix Klein as "Kleinsche Fläche" (Klein surface), and only later obtained its present name due to translation mistakes.
Showcase of this model is Case number 28
Hilbert, D.; Cohn-Vossen(1932). Anschauliche Geometrie, Springer-Verlag, Berlin, p. 271 f.. Online version