Most cameras have a maximum shutter speed of about 30 seconds - more than long enough for the majority of photography needs - but in some instances, such as night time or star trails photography this is not long enough to get enough light onto the sensor to capture the picture. To accommodate the need for longer exposures, manufacturers provide the Bulb setting, usually identified by a B on the settings dial.
Long exposures are wonderful for low-light photographs, or where you want to create a sense of movement (for example if you're photographing water or car light trails).
Camera shake is a potential pitfall of using the bulb exposure. If you physically keep the camera shutter button depressed, it's very easy to nudge the camera, even when you're using a tripod. This means that even the parts of the photograph you want to be static will look out of focus. If you need to keep the camera steady, it's best to use a remote shutter release - available either as a wired connection to the camera, or as an infrared remote. Some wired releases can be programmed to keep shutters open for longer exposures with the camera set to Bulb.