Bokeh describes the visual quality of the out-of-focus areas of a photograph.
Quick link for the common "how to" question: How can I maximize the "blurry background, sharp subject" (bokeh) effect?
Depth of Field (DOF) is an important part of the artistry of photography. When used as an artistic device (as opposed to merely focusing incorrectly), a DOF that doesn't include everything in a photograph will result in parts of the photograph appearing in-focus (sharp) and other parts appearing out-of-focus (blurry).
Although it is common to refer to the out-of-focus portions of the photo as bokeh, this is not strictly correct. The term "bokeh" actually refers to the quality of those out-of-focus areas; specifically, the nature of the blurring. Bokeh is generally considered favorable if blurring is smooth and free of artifacts introduced by the lens (glass or aperture blades). Less desirable bokeh will show geometric patterns or other optical imperfections; these are generally introduced by the lens rather than the camera body
Bokeh is heavily influenced by aperture settings, so in addition the amount of blur, the quality of blur (this is bokeh!) will vary for a given lens as its aperture varies. It is therefore possible for a lens to produce the most blur as its widest aperture, but its most pleasing blur at a smaller aperture.