If you have a video camcorder, then you are used to video and sound getting recorded on the same tape. The sound and video are always synchronized because they are recorded together in one place (See How VCRs Work for details).
When filming a movie, the pictures and sound are recorded separately. The picture is recorded on film in the camera, and the sound gets recorded on a separate analog magnetic tape recorder (or, more recently, on digital tape such as a DAT tape). Because they are recorded on two different devices, you need a way to synchronize them.
A clapperboard -- the black and white board you are talking about-- is the traditional way to handle the synchronization. The bottom of the clapperboard is normally a slate of some sort on which you can write the scene and take number. This information helps identify the shot during editing. Once the tape recorder and camera are rolling, the clapperboard operator places the clapperboard in front of the camera so the camera can see it, reads the scene and take information so the tape recorder can hear it, and then claps the clapper. During editing, it is easy to synchronize the visual of the clapper clapping with the "clap" sound it makes on the tape.
The digital slate is the more modern form of the clapperboard. The tape recorder contains a timecode generator. The timecode is recorded continuously on a special track on the tape, and the timecode is also displayed continuously on a large LED display on the digital slate. By showing the digital slate to the camera before the action starts, the editor knows exactly what the tape's timecode is and can synchronize it with the film. Sometimes a digital slate contains a clapper, but generally it is not needed. The digital slate normally has a slate area, because identifying each shot is still extremely important during editing.
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