What does the f in f-stop stand for?

The f in f-stop stands for "focal length". Focal length is the length of a lens (18mm, 50mm, 200mm, etc.)

The notation f/4 is a shorthand notation for the diameter of a lens opening, where focal length is divided by 4. F-stops show a ratio of the relationship of focal length to lens opening.

Here is Ansel Adam's explanation, from his book, The Camera:
“The lens aperture is simply the diameter of the lens opening, expressed as a fraction of its focal length. Thus a lens of 4-inch focal length with a diameter of one inch has a relative aperture of 4/1, or 4. The aperture designation is expressed as f/4, indicating that the aperture is the focal length/4. Another 4-inch lens that has a diameter of 1/2 inch would be an f/8 lens.”
Adams, Ansel. The Camera. Boston: New York Graphic Society, 1980: 46.

Some examples:

A 200mm lens whose aperture diameter is 50mm has an f-stop of f/4
200/50 = 4

A 200mm lens whose aperture diameter is 25mm has an f-stop of f/8
200/25 = 8

An 18mm lens whose aperture diameter is 4.5mm has an f-stop of f/4
18/4.5 = 4

All f/4 lenses let in the same amount of light, because the ratio of focal length/diameter is the constant.