Depth of field is one of the most useful creative controls on any camera.
It enables you to see how any given aperture setting will affect how much of your photographic scene will be in sharp focus. Aperture settings don't just affect how much light enters the lens, they determine how much of the scene in front of and behind the subject which you've focussed on will also be in focus. The distance between the nearest object in sharp focus and the most distant is called the depth of field.
Wide open apertures (i.e. lowest numbers) give you the shallowest depth of field and vice-versa.
Modern cameras always show the image in the viewfinder or LCD using the lens aperture wide open, regardless of what you've actually set: this allows maximum light into the lens to allow you to clearly see the scene and the lens only close down to the correct aperture at the moment that you press the shutter. The depth of field button (more correctly called the depth of field preview button) enables you to close down the aperture to what it's actually been set to so that you can see exactly what is in sharp focus; when you press it the scene will darken as there will be less light entering the camera, but if you look at a foreground or background subject which is out of focus before you press the button you'll notice that it becomes sharper when you activate the preview. The button will not have any effect at all if you have the lens set to it's maximum (lowest number) aperture, as the aperture that you're viewing the scene at is identical to the one you're taking the photo at.
Understanding depth of field and how you can manipulate it is vital to taking stunning photos:-
Say you want to take a photo of a bee on a flower: if you leave the camera set to auto, or select a medium to small aperture then the photo will show the bee, the flower, and everything in front and behind making a confusing and busy shot. If you select a wide open aperture then the bee will be in sharp focus (if you're really close, maybe only it's head), the flower, or parts of it will be in sharp focus, and the foreground and background will blur out making the bee and the flower the most important compositional elements in your shot.
Alternatively, you may be in a situation where you need to lift your camera quickly and take a shot without disturbing the subject. You don't know exactly how far away your subject will be, but you know it will be between, say, five feet and twenty feet. If you use your camera as normal, you'll see the shot, lift the camera to your eye, wait for focus (if using an autofocus camera, it might not even focus on what you intend). By the time the shutter has activated the moment has passed or the subject has seen or heard you and gone. Using depth of field you can manually prefocus to a point about a third of the way into your d.o.f. (in this case, ten feet) and select the correct aperture to give you a fifteen foot d.o.f. The setting varies with the lens, but you'll almost always get away with f8). When you see the right shot you just lift the camera and fire without worrying about focus and if you've done so correctly your subject will be sharply focussed. Of course, you could set the lens to minimum aperture, but this can result in the shutter speed being too low for the light conditions and causing unsharpness due to movement of the subject or your camera.
The technique is known as hyperfocal focussing and it explains why some lenses have various markings on them in various colours with aperture numbers next to them, they're a simple depth of field calculator for any given aperture setting. I'd provide a link but it's better if you search yourself as some sites go into what may be far too much detail about the subject.
Hope this has helped you, all that I ask in return is that you take a moment to rate my answer. If there's anything which you want me to clarify further then add a comment to my answer and I'll return as soon as I can to assist you some more.
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