The film plane is the plane in which the film (or the electronic sensor) is placed. In a properly-designed camera, this is synonymous with the focal plane, onto which light entering a lens is focused. In many cameras with simple lenses, the film plane is not a plane (i.e. not flat), but curved, to compensate for inadequacies of the lens, and in swing-lens panorama cameras, curved to follow the movement of the lens.
The lens aberations are spherical, not cylindrical. Such a curved film plane would need to be cylindrical and would then compensate on the x axis but not on the y axis. Some camera makes did it as a compromise to have the smallest possible camera.