This article is from the Photographic Lenses FAQ, by David Jacobson with numerous contributions by others.
Vignetting is a reduction in light falling on the film far from the center of the image that is caused by physical obstructions. Light falloff is a reduction of light far from the center because of fundamental optical reasons: First, an off-axis object sees a foreshortened apparent aperture (entrance pupil) so less light is collected. This results in a cos(theta) falloff, where theta is the angle off axis. Second, in a rectilinear lens the solid-angle-to-area magnification increases with cos^3(theta), spreading the light from a patch near the edge over more film than if the patch had been near the center. (The patch is presumed to face the camera at a constant very large distance.) As a result there is an overall cos^4(theta) falloff. The optical designer can compensate for these effects by making the entrance pupil enlarge and tip when viewed from off the optical axis. An alternative approach is to compensate by using a filter whose density varies appropriately with distance from the center.