In the picture below we have low depth of field. See how the boy in the front is in focus, but everything else is out of focus? This is great when you want to draw attention to just one subject in your image, for example the boy in this picture. I wanted him to be the focus. (Also notice the rule of thirds!)
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The picture below has great depth of field. Everything in the picture is in focus. This is great for scenery if you want the whole scene to be the focus of the picture. It also is important to use this technique when taking pictures of large groups of people so that everyone in the picture is in focus, and not just one person is the center of attention.
You can get low depth of field with your point and shoot camera by focusing your camera (hold the shutter release half way down) on the subject you want to be in focus then moving it away to capture the whole scene. The camera should stay focused on your subject. Also try putting it on the macro setting. With SLR cameras make sure you have your aperture set at something like f/5.6 or smaller like f/2.8.
I love working with depth of field because it brings a whole new dimension to your images! Have fun playing with it!