The f-number is a value that represents the amount of light passing through the diaphragm when the lens is focused at infinity. The f is an abbreviation for the term factor, and describes the mathematical ratio of the focal length of the lens divided by the diameter of the aperture.
The f number value labeled on the lens is accurate only when the lens is focused at infinity. As the lens moves forward from the infinity position, to the focus on objects that are closer to the lens, the image distance becomes greater than that noted on the aperture ring. The differences is within the exposure latitude of the film for all but near objects, and taken care of automatically with TTL metering when the lens-to-film distance becomes substantially greater than the focal length. Then, additional exposure compensation must be made.
Modern aperture rings are labeled in the series f/1, 1.4, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45 and so on. The numbers may appear as random in order, yet each is the previous number multiplied by the square root of two (1.4). A number is thus doubled every other number. Each numerical stop higher in the progresssion allows one-half the amount of light to reach the camera as the f-number preceding it.
The f-number is also used to describe the relative speed of a lens. The lower the f-number, the faster the lens. An f/1.4 lens may be described as faster than an f/2 lens in that twice as much light passes through the f1.4 lens in a given amount of time.
The setting of an f-number on the lens affects more than just the quantity of light passing through it. When stopping down the lens (using a higher f-number) there is an increase in the depth of field: the small the aperture the greater the depth of field. And although apparent sharpness is increased, this is true only up to a point. Further stopping down the lens will subject the ray of light to more diffraction at the aperture opening.
Source: Nikon School Handbook