Depth of field is one area that I find people always are asking about so here are a couple pictures to help understand it a little better.
The lower/smaller the aperture
Smaller the number that you use, the shorter/narrower the depth of field. So in the first picture I shot this using an aperture of f5.6. Notice how the subject (the fence & wire) are sharp and the background is out of focus? Great to blur the background and to focus a viewers attention on a model or a bride.
In this picture I stood in the same spot but I changed the aperture to F10. Notice how everything is in focus? This is good for groups of people or subjects that are not lined up in a perfect row. This can also cause the picture to look confusing as you do not know what the subject is. If not sure try a wide and a narrow.
Using a short/narrow depth of field allows the photographer to separate the subject from the background. In a busy scene like the above images, this allows you to see the subject better. You are not be bothered by the busy background. If you really want the effect to be dramatic then you need to get a lens that goes down to F1.8 or F2. That way you will really see the effect.
Having your focus a little off with a short depth of field will show in your subject being out of focus. A longer depth of field can cover for bad focusing, a short depth of field will show bad focusing. As mentioned before if in doubt try both.