What does a camera's maximum flash sync speed mean?

With mechanical shutters on most common DSLRs, they actually consist of two shutter curtains, that move in the same direction.

At slow shutter speeds, one will open, and then, after some time, the other will close. As these are mechanical devices, there is a maximum speed at which they can reliably move. With really fast shutter speeds only a narrow band is exposed to the image through the lens at any one time as the second curtain begins to close before the first curtain has fully opened. The faster the shutter speed, the narrower the opening between the two curtains as the second one chases the first one across the focal plane. The transit time of each shutter curtain is the same for all available shutter speeds in most modern cameras. It is the time interval between the movements of each that determine the exposure time. Even though each point on the sensor may only be exposed for as short a time as 1/8000 of a second, it still takes the curtains around 1/400-1/200 second (depending on the camera's design) to move across the entire sensor and take the picture.

The sync speed is the fastest speed at which the entire sensor is exposed to the light through the lens at the same time.

High Speed Sync is available on some flash/camera combinations which allow flash to be used at faster shutter speeds. This is achieved by strobing the flash to coincide with distinct bands of the sensor, such that the entire picture is correctly exposed. As an example, if the shutter speed were to allow at most half the sensor to be visible, the flash would strobe twice - once for the top half, and once for the bottom.