This article is from the Photographic Lenses FAQ, by David Jacobson with numerous contributions by others.
The focal length of the lens divided by the diameter of the aperture (as seen from the front). It is also called an f-number, and is written like f/8, which means the aperture diameter is 1/8th the focal length.
The term is used both in regard to the maximum aperture of a lens and in regard to the aperture selected in a specific situation.
The brightness of the image on the film is inversely proportional to the f-number squared. The depth of field increases but diffraction is worsened when using a large f-number. The effective f-number for all 3 effects changes if the lens is focused extremely close. See Q7.
The term "stops" purportedly comes from old technology in which the aperture was selected by turning a wheel with various sized holes in it, each one of which let in twice the light of the preceding one. Thus the phrase "open up a N stops" means to change to an aperture allowing in 2^N times as much light, and conversely with "stop down N stops".