Aperture: The Hole Story
One of the most important parts of your camera is something that isn't there at all. The aperture is basically just a hole through which light reaches your sensor. But that's only the beginning of the story.
At the heart of your camera is a gaping maw. A black hole that opens to allow the outside world to enter your camera and molest the sensor within. Don't worry. That's the say it's supposed to work. A camera wouldn't be much of a camera without such an opening. And it wouldn't be much of a camera if that outside world didn't leave its mark on the sensor in the form of counts in each of its many photosites (pixels). But not all openings are created equal. And therein lies our tale.
As light enters a lens, it's obviously constrained by the diameter of the front element as well as that of other internal lens elements and mountings. But unless you are shooting with a pinhole camera lacks any other means of aperture control, the lens opening is rarely the determining factor. The camera aperture control adjusts the diameter of the opening in an internal diaphragm that exists specifically for this purpose. In the early days of photography, practitioners of the art had to insert one of a set of premade boards with different diameter holes in them in order to select the aperture. As it became increasingly possible to manufacture complicated but reliable alternatives, cameras started to be fitted with adjustable aperture diaphragms built in. With just a twiddle of a dial (or rotation of a lens ring if you have any older lenses), you can now change the aperture on the fly. The lens elements may constrain the widest aperture available, but the aperture diaphragm opening is what lets you pick anything up to that.
If you compare the sizes of various lenses, you've no doubt noticed that longer focal length lenses are not only longer, they also tend to have larger in diameters than shorter focal length lenses with the same maximum aperture. You can thank physics and a bit of history for this, but the explanation lies in how those funny "f-stop" numbers are defined. Rather than merely being a measure of aperture size, f-numbers are actually a ratio of the opening diameter to the focal length. As light gets focused through the lens and projected toward the sensor beyond, the image spreads and becomes dimmer. Brightness decreases as focal length increases, so to keep the effective brightness the same, you need a bigger hole to begin with to compensate. Defining f-stop numbers as this ratio allows us to compare effective brightness independent of focal length.
As to why the aperture numbers in the standard f-stop series are such seemingly random values, you can thank geometry and yes, a bit of history. The amount of light that can get through the aperture increases in proportion to the area of the opening rather than the diameter or radius. If recalling memories of math classes from when you were a kid doesn't make you go run shrieking, you probably know that the area of a circle is determined by the constant pi multiplied by square of the radius (r). Never mind how many decimal places of pi you can remember, the important thing here is that doubling the area only changes the radius by the square root of two. That fact yields the number sequence where each successive value differs from its neighbors by the square root of two. And that means we all have to live with 1.4, 2, 2.8, 4, 5.6, 8, 11 and so on. See, they told you in school that math would prove useful in life.
While pi are squared, aperture are round. Or at least they are as round as lens makers can make them. Mechanically, the changing aperture is formed by means of interleaved aperture blades that push in from all sides. To make a closer approximation of a round opening, you need more aperture blades, each one shaving off a smaller portion of what needs to be cut off. The art of how the iris gets shaped and how the blades interact to do so creates the ephemeral and mysterious out of focus blurring known as "bokeh." More blades create softer bokeh. Odd numbers of blades minimize interactions from blades on the opposite side of the opening creating softer bokeh. For some photographers, the quest for good bokeh falls somewhere between magic and religion. Others are puzzled at what all the hubbub is about. Most of us fall somewhere in between.
Aperture has a direct effect on depth of field, the range of distance from foreground to background within an image that appears to be in focus. A small aperture opening will yield a greater depth of field, and a larger one will cause resulting images to have a shallower depth of field. Be aware that, by default, traditional camera viewfinders won't show you how depth of field changes as you adjust the aperture. In order not to darken the viewfinder image, the lens is designed to only stop down to the selected aperture once you press the shutter release. You have to use the depth of field preview button, or on a more modern camera, the live view display or electronic viewfinder. If all these seem too complicated, you can always fire off a shot and check out the results afterwards on the camera back LCD. But however you may approach it, aperture selection matters, and your images will thank you for giving due attention to the task.
Naturally, there are limits. Stop down the aperture completely, and no light can get through. But even short of that, diffraction can increasingly become a problem from small apertures. Light rays passing through the center of a typical aperture opening find an unobstructed route on their way to the sensor. Rays passing too near an edge though can become slightly bent from diffraction. As the opening gets smaller and the edges close in, the portion of light that must pass near an edge increases. Well before the hole has closed down completely, there simply won't be any clear paths through the opening. Everything ends up being near an edge. There's a reason that aperture number series typically tops out at f/22 or perhaps f/32. And if you're shooting with a DX or similarly sized sensor that is less than "full frame," diffraction can become a problem even before you reach that limit since the recorded image has to subsequently enlarged more to reach a given print size when starting from a smaller sensor. For a similar reason, medium format cameras with their even larger sensors often support lenses they go all the way down to f/45. It's not uncommon for photographers to crank the aperture all the way to the smallest opening without much thought in order to maximize depth of field. To avoid problems from diffraction, a better approach is to only stop down as far as necessary for a given situation.
Together with shutter speed and ISO, aperture is one of three variables that determine exposure. The old-school way of conceptualizing this was that ISO determined how sensitive the recording medium was, and that aperture and shutter speed together controlled how much light was let in in a reciprocal relationship. Once you loaded the film, ISO was predetermined. From there, you could either let in a lot of light (wide aperture) for a short period of time (shutter speed), or a small amount of light over a longer period of time to end up with the same total exposure. Decisions made could naturally have other impacts, but exposure could be kept the same by changing one by the inverse of any changes made to the other. This way of thinking is a holdover from the film days when you had to commit to a certain ISO for an entire roll, leaving you with just two remaining variables to adjust.
Now that digital cameras have improved to the point where higher ISO settings don't necessarily lead to noisy images, we can more freely make use of all three exposure variables for every shot. Depth of field can be chosen via the choice of aperture, the desired rendering of motion can be dialed in by selecting the shutter speed, and then exposure can be compensated for via the ISO. Rather than being the first exposure variable chosen, ISO is more properly placed third in line for adjustment. So long as ISO doesn't have to be pushed too far, this gives us greater freedom when selecting aperture and shutter speed for their creative potentials independent on their exposure influences.
I should also note that using a teleconverter effects the aperture. Adding a 2x teleconverter doubles your focal length but does nothing to alter the lens diameter. The ratio between diameter to focal length thereby decreases by half. Yes, again with the math. Sorry about that.