Depth of Field, in my simple, lay view, is how much of a photographic image is in "sharp" focus (a relative term) from front to back. The optics of a lens limits this.
Generally, two "adjustments" on a DSLR/SLR body/lens combination will affect DOF. The first is the focal length of the lens, and the second is the size of the opening (aperture). The following are generalities (there are specially designed lenses and a more scientific approach to all of this):
Longer focal length lenses at the same distance, have less inherent DOF. This means that the part of the image which is in sharp focus from front to back is much more narrow. Where that "in-focus" part of the image is depends on where in the image, the lens is actually focused.
Shorter lenses (especially wide-angle) have much more depth of field. The practical impact of this is that with a longer lens, you can get areas -- intentional or not that are out of focus. When you are trying to isolate a subject from its background (or sometimes foreground), this out of focus effect can be pleasing, particularly with a "busy" background. But if you aren't careful, you can also get parts of the image that should be in focus out of focus.
Likewise, a smaller aperture yields greater DOF than a wider one.
I am not certain how much control you have with that on the P&S camera --- but generally not nearly as much.
The "settings" on the camera are usually simply the manufacturer's combination of the above. But then you have the camera doing the "thinking" rather than yourself. Though they are pretty "smart" they are not near as good, in my view, as you are at choosing which of these elements work best in an image.
If you are going to shoot a small aperture with a longish lens in good light conditions for this stuff, you are undoubtedly going to deal with slow shutter speeds. This is why it is critical to have a tripod (and I believe, remote release).
Some will suggest that the self-timer on the camera is just as good -- but there are some problems with it.
You will also have to work with wind and possibly, (on cloudy days) moving light conditions. Its hard to time a let up in wind with the self-timer. If light conditions are rapidly changing, its hard to time them, too.
For a much better organized and readable explanation of DSLR DOF, aperture and shutter speed relationships, you can read the tutorial on my LIGHTCENTRIC PHOTOGRAPHY BLOG, or purchase Bryan Peterson's "Understanding Exposure" (also on my Blog Bookstore). The latter is well worth the few $ and the read. My tutorial? Well, you be the judge
The photos I took that started this discussion were close ups of flowers. My preference for this is a blurry background. After Andy's post I did try other f-stop settings. I've had fun taking all sorts of photos at different settings, but I'm giving my hard drive a nervous breakdown so I've had to add an external hard drive. I guess the next thing to get is better software, like photoshop, but first I need a new laptop, because this old bugger can not handle photoshop or even elements.