Why use a clapperboard?

Clapperboards serve two main functions, both of which make an editor's life easier.

Synchronisation

The "clapper" part of a clapperboard is used to synchronise the audio with the pictures.  The editor looks out for the precise moment of the strike in the footage and lines this up with the clear spike in the audio waveform and voilà! - perfectly synced pictures and sound.

The clapperboard is enjoying a comeback of late because of the massive popularity of DSLRs for video.  DSLR cameras do have onboard audio, but it is a bit of an afterthought and there are various technical and practical issues with recording sound straight onto your camera.  One of the best ways round this problem is to use a separate audio recorder, such as the Zoom H4n or the Tascam DR 07 (pictured) which have good quality mics in a stereo pair and two XLR inputs.

When filming with a DSLR and a separate audio, you can use the low-quality audio from the camera as a guide to get the high-quality audio from your audio recorder in sync with the pictures.  You just look for the spikes in both tracks and line them up.

The advantage of using a clapper to create your sync-point spike is that you get a nice, clear beginning to the sound that's easy to spot in the waveforms.

Slating

The other function - the "board" part - is called slating, and this can save you a huge amount of time in the edit.  This video from Framelines TV shows you everything you need to know about using a clapperboard, including what to write on the board.