Lens Aperture
The aperture of a lens determines the amount of light that passes through the lens to the film or digital sensor. It's a diaphragm of overlapping leaves that is either controlled by an external ring on older lenses or set by controls on the newer cameras which sets the lens electronically.
In the case of older cameras and lenses, turning the ring one way opens up the lens so there's no obstruction to incoming light. Turning it the other way closes down the lens until only a small circular opening is left for the light to pass through.

With newer cameras, buttons or a dial on the camera will let you set the appropriate aperture. However, unless there's a depth of field preview button, you may not see any change. The camera will close the lens down to the value you've set just before it takes a photo.
f-Stops
On all lenses, aperture is calibrated using what are called f-stops (in fact, the terms are almost interchangable). Along with the focal length of the lens, the maximum aperture is usually used to specify a lens. By that I mean that if the lens is described as an f/2.8 or an f/4 lens, for instance. The actual range of f-stops varies between lenses but the sequence is always the same:
f/1, f/1.4, f/2, f/2.8, f/4, f/5, f/5.6, f/8, f/11, f/16, f/22, f/32, f/45, f/64
Maximum apertures can be "non-standard" - that is, not in the above list, so it's common to see short focal length zoom lenses with wide open settings of f/3.5. Longer focal length zooms may have maximum values of f/4.5.
Standard lenses, sometimes known as "fixed" lenses (i.e. they don't zoom), tend to have higher maximum f-stops than zoom lenses. On film SLRs (Single Lens Reflex), the standard 50mm lens that comes with the camera will have a wide open setting of f/2, f/1.8 or even f/1.4.
Regardless of the focal length of the lens or the diameter of the lens, f-stops denote relative apertures. - they actually specify the ratio of the diameter of the aperture to the focal length of the lens.
Let's take an example. On a 135mm telephoto lens (good for isolating something in the distance), an aperture setting of f/4 means that the physical diameter of the diaphragm opening is 33.75mm. How did I get this figure? Divide the focal length of the lens by the f-stop. So, 135 / 4 = 33.75.
On a 28mm lens (great for wide landscape photos) with its aperture set to f/4, the size of the opening will be 7.5mm (28 / 4 = 7.5).
That's Neat!
If you're mathematically minded, you probably noticed that each successive f-stop is the square-root-of-2 (approx. 1.414) times the previous f-stop.
Decreasing your f-stops double your exposure. For example, if your exposure is 1/125 sec at f/5.6 and you change your aperture to f/8 need to double your shutter speed to 1/60 sec. If you changed to f/8, you'd need to double your shutter speed again to 1/30 sec.
Likewise, increasing your f-stops halves your exposure for each f-stop. So, changing from f/4 to f/2.8 would mean halving your shutter speed from 1/125 sec to 1/250 sec.
Simple Rules
Why is Aperture Important?
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