It's a technique that adds artfulness to portraits, close-ups, and other shots where the background doesn't need to be in focus. Once you recognize it, you'll start seeing the bokeh effect everywhere.
Bokeh, pronounced BOH-kay, derives from the Japanese word boke, which means blur or haze or boke-aji, which means blur quality. This quality is caused by a narrow depth of field, the distance between the nearest object in focus, and the farthest.
When using a DSLR or film camera, a combination of the aperture, focal length, and the distance between the photographer and the subject creates this effect. Aperture controls how much light is let in, while focal length determines how much of a scene a camera captures, and is expressed in millimeters (i.e., 35mm).