Depth of field and bokeh are two confusing terms in photography. What is depth of field on a flat image? And bokeh sounds like a misspelling.

Depth of field
Depth of field is the range of distance in the image where objects are sharp and in focus. A large DOF means more of the image is in focus and therefore sharp, while a small DOF means only a small portion of the picture is sharply in focus while the background and foreground is blurred and fuzzed out. While it seems logical to always aim for more sharpness in your photographs, the truth is camera lenses can only really focus at one distance, but either the fuzzy blur is so gradual as to be imperceptible, or it's obvious in contrast to the sharp area of the photo.

For photographs like landscapes, no one wants half the image to be blurry fuzz, but for other photographs like portraits or still life, a little fuzz is a great way of directing the gaze to the subject of the photograph.

Aperture and DOF
The interesting thing about Aperture is the smaller the size of the aperture (so bigger the f number), the clearer and crisper the image will be. So if you want to get a small DOF, lower the aperture. It's important to note that the amount of defocusing also depends on the distance of the subject and the surrounding area. DSLRs are pretty useful since they have focus points that you can use to set where the camera will focus, otherwise you can point the centre of the camera at your subject, lock the focusing and then shift away to recompose before pressing the shutter. That way you will have your subject clear on one side of the photograph while everything else is nice and fuzzy.

Bokeh
Bokeh is a Japanese term, which literally means fuzzy. DOF signifies the part of the image that is in focus, the depth of field in the image that is clear, Bokeh signifies the part that is unfocused. The reason people have given it a name, is because the fuzz can be good or bad. Good bokeh is nice uniform fuzz that looks nice, while bad bokeh just looks messy. The quality of the fuzz depends on the lens. Sometimes bokeh is clearly defined as little circular fuzzy shapes; a lens with few diaphragm blades might give less circular and more angular shapes.

DOF and Bokeh go hand in hand together, the only thing you as a photographer needs to decide is how much of the image is going to be blurred and where you want the image to be focused, or if you want any blur at all.