Aperture
The aperture is the part of the lens that opens to let light into the camera. Also known as the "f-stop," the aperture setting can vary in size, with subsequently larger openings (smaller f-stop numbers) letting in more light than smaller openings (which, confusingly enough, have higher f-stop numbers).
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Consider the fountain images again. In order to capture the first one at 1/13th of a second, the aperture was stopped down to f22, the smallest opening possible for the lens used. The bottom image, captured at 1/1000th of a second, was shot at f2.2. Notice the crisp background in the first and the blurred background of the second. A small f-stop limits how much of the image is in focus and is great for portraits. A larger f-stop is good for group shots, like capturing a crowd at an event, or taking a picture of an entire room.
Depth of Field
Depth-of-field is an expression of how much of a photograph, from front to back, is in focus. Changing the aperture can result in two different images of the same scene:

The image on the left has a very narrow depth of field; only the flower's petals are in perfect focus, as is a relatively small length of the tape measure. When the lens is stopped down to the smallest aperture, f22, the entire tape measure is in focus, as is the flower.
Setting a camera to capture a narrow depth of field is particularly useful in isolating the subject of a photo against its background, as in this photo of my fellow TechSource blogger, Jason Griffey:
Narrow depth of field is ideal for portraits, whereas wide depth of field is required to capture details in a landscape or interior, although narrow depth of field can be used effectively in architectural photography as well:
(Left: f20; Right: f3.2)
As mentioned above, changing the aperture changes the amount of light that it let into the lens. The shutter speed must be changed accordingly to compensate. Most of us rely on our cameras to make these adjustments for us, but here is where knowing a bit about how your camera works can help you dictate what sort of photos you get, instead of the other way around. For a more in-depth explanation of depth of field and its concomitant terminology, visit BernieCode.
Note also that the lion's face in the photo on the left is slightly elongated. I haven't yet researched why this is, but I suspect it has to do with differing focal lengths. These photos were taken with the same lens.
Applying these principles in your library
In a library setting, a higher shutter speed would let library staff capture fast-moving toddlers at storytime, while a slow shutter speed (and sitting the camera on a tripod or other stationery object) would make for a great night shot of the building.
I already mentioned that a smaller f-stop makes for great portraits or other photos where it's necessary to isolate the subject from the background. A larger f-stop is not only great for group shots but for taking photos inside or outside the library building, for brochures or websites. We have exciting events and beautiful buildings; show them off with pictures!
One of the biggest advantages of digital cameras over film cameras is that it costs little-to-nothing to take dozens, even hundreds, of shots. Experiment with your camera by shooting the same scene, changing one setting at a time. Any digital camera will have different modes that allow the photographer to fix one value while varying another; it's a great way to learn.
Up next: Cameras and modes explained
About the "Take Pictures, Tell Stories" series