Boy, this thread has gotten confusing really quickly. Let me confuse you even more.
Depth of field: area of your photo that is in sharp focus. If you focus on your subject the DOF will extend in front of and behind the point you are focused on. In most cameras the DOF is parallel to the film plane. With view cameras and shift lenses you can alter the position of DOF in relationship to the film plane. This would allow what appears to be a greater DOF (it is actually just a repositioning of the DOF, you are using the height or width of the DOF rather than the depth (HOF? WOF??)). This is useful in close focusing situations.
Besides aperture, DOF is also affected by lens focal length and distance to subject. Wide angle lenses have greater DOF than telephoto lenses at a given aperture; the DOF at f/8 on a 28mm lens is huge (measured in feet), while the DOF at /8 on a 200mm lens is pretty small (measured in inches). The closer the lens is focused the shorter the DOF. With the lens focused at 20 feet f/16 is very deep; focused at 3.5 feet f/16 is very shallow.
Aperture: the hole in the lens. Controls exposure and DOF. The smaller the aperture, the greater the DOF. A non-photographic example of this is folks who need glasses for night driving: in the daytime when it's bright their pupils close down (small aperture/greater DOF), at night when it is dark their pupils open up (large aperture/less depth of field). Aperture size is refered to using f/number.
The f/# is arrived at by dividing the focal length by the actual aperture size. Example: a lens with a focal length of 100mm is at f/2 when the aperture is 50mm wide, it is at f/4 when the aperture is at 25mm wide, it would be at f/22 when the aperture is set to 4.5mm wide. So the larger the f/#, the smaller the actual aperture size, and the greater the DOF.
Old school lenses went from widest to smallest: f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32, f/45, f/64.... Each number was approximately one stop (a halving or doubling of the amount of light). This standard helped the photographer doing math in the field. Auto-focus SLRs have built in computers and can do the everchanging math of the big zoom lenses so you will often see non-standard f/#s in the viewfinder such as f/9 or f/13.
Modern lenses are designed to stay wide open while you are looking through the viewfinder composing and focusing. When the shutter button is depressed the aperture closes down to the selected size a split second before the shutter opens, and returns to wide open when the shutter closes. Some cameras have a DOF preview feature. This allows the photographer to close down the aperture to the selected size, and DOF can be assesed through the viewfinder.
Fixed focal length lenses used to always come with a built in DOF scale on the focusing ring so that even without DOF preview the photographer knew their DOF. Zoom lenses would require something akin to a slide rule to figure it for all the possible combinations of focal lengths, aperture sizes, and distance to subject. Fortunately you can find websites that will show you how to build this device (search for "depth of field calculator").
While you're at it, search for "hyperfocus".