In the nomenclature of photography, the lens aperture is its diameter. We adjust the working diameter to regulate how much light energy the lens passes during the exposure.
As you know, the focal length of the lens expresses the power of the lens to magnify. When we zoom we are changing the focal length. We do this to obtain more magnification or to acquire a wide-angle view. Thus, as we zoom, the size of the image of objects will expand or contract. As we zoom to obtain more magnification, we are forcing the image-forming rays to spread out. Thus the image dims. Conversely as we zoom to gain a wide-angle view, the image-forming rays contract and the image brightens.
The changing image brightness that comes with the zoom must be skillfully neutralized, otherwise over or under exposure will be the result. One might create a mechanical linkage that adjusts the diameter of the aperture. In this way, the exposure would be organized to stay constant throughout the zoom. This has been tried, but such a mechanism adds too much cost.
Best is an optical solution. The diameter of the aperture can be made to perform as if it were a variable by placing it behind an assemblage of lenses that magnify. The assemblage moves with the zoom; thus they change the apparent diameter of the iris with the zoom. This works to keep the exposure constant. It takes skillful lens design, and this adds to the cost of zoom lens. A very expensive zoom will likely maintain a constant exposure throughout the zoom. However, most moderately priced zooms give up the ghost as the magnification increases. These will not maintain a constant f-number. Thus they top out and the f-number at max magnification is about two f-numbers stopped down.
The relationship: The capability of the lens as to image brightness intertwines the capture area (working area of the circular lens) and the focal length. We use a ratio to express this as a single value. We divide the working focal length by the working iris (aperture) diameter to obtain the “focal ratio”. In the jargon of optics this is called the f-number. The f-number is interwoven in the geometry of circles. If you multiply or divide the diameter of any circle by the square root of 2 = 1.4, you have calculated a revised circle with double or half the surface area. Thus the number set: 1 – 1.4 – 2 – 2.8 – 4 – 5.6 – 8 – 11 – 16 – 22 – 32. Each value going left increases image brightness 2X, going right, image brightness is cut in half.. The beauty is; this works regardless of the focal length or iris diameter. In other words, any lens set to f/4 delivers the same image brightness as another lens set to f/4 regardless bigness or littleness of the power or working diameter.