DoF
Depth of field (DoF) relates to how parts of a photograph will appear in or out of focus. When we see things in real life, our eyes automatically refocus at whatever we happen to be looking at. Everything appears in focus because of the fast focusing. With a camera lens, it does not work quite that way. The lens is focused at a single distance. Everything else is not in perfect focus; however, something slightly ahead or behind the focus point will still look in focus. Objects further ahead of the focus point or further behind it will gradually become less focused and more and more blurred. The points at which most people would call the objects still in pretty good focus is referred to as the depth of field.
The focusing distance, the lens focal length, the aperture/f-stop, and the digital sensor all play together to determine the depth of field. Smaller point and shoot cameras have a far greater depth of field than larger format cameras. This is sometimes a good thing, and sometimes not so good. You can blur out distractions in the background with a limited depth of field, or you can help place the subject in her environment with a greater depth of field.
The simple rule for any camera is a larger aperture (smaller f-stop number) will have a shallower depth of field. A smaller aperture (large f-stop) will have a greater depth of field. Also, the closer you are to the focus point, the shallower the depth of field will be. Standing in the same spot, a wide angle lens will appear to have a greater depth of field than a telephoto lens.
Examples
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Merci is within the DoF but the tree starts out of fous, goes in focus under Merci, and falls out of focus in the background
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Sienna has her right nipple in perfect focus, but her left nipple is outside of the depth of field
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BiKate is within the DoF but the putting green grass starts out of focus, goes into focus around BiKate, and falls out of focus beyond her
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Raluca is the focus point, and each stack of library books falls further and further out of focus
Sometimes you might like to have a big depth of field to be able to determine what is in the background, and other times, you wish the background to fall out of focus to highlight the subject.
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Sienna is in sharp focus, and the background remains in near enough focus for some details to be seen
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Sienna is again in focus, but by limiting the depth of field she pops out of the scene leaving the background and foreground blurred
Details
A more detailed explanation of depth of field includes topics to be searched on Google: "circle of confusion", "airy disk", and "resolution." The old guides to depth of field used a standard of printing to an 8x10 and being able to distinguish the in or out of focus points. So what a depth of calculator might tell you is in focus, may not seem so to you depending on how you are viewing the image.
Take your camera off automatic and experiment with the exposure dials! A faster shutter speed will require a larger aperture (smaller f-stop) and result in a shallower depth of field. A faster ISO setting can either increase the shutter speed for the same depth of field or allow a smaller aperture for a greater depth of field.