When referring to flash exposure, what does "max sync speed" mean?
With mechanical shutters on most common DSLRs, they actually consist of two shutters, 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 move, and with really fast shutter speeds only a narrow band is exposed to the image through the lens.
The sync speed is the fastest speed at which the entire sensor is exposed to the light through the lens.
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.
It refers to the maximum shutter speed where both the sensor is fully exposed to the light. Shutters consist of a rear and front curtain. For shutter speeds faster than that, the rear curtain trails increasingly close behind the front shutter before it reaches the other end.
One benefit of sync speed that has nothing to do with flash is the amount of distortion that you will get with very fast moving subjects. It is the same as the rolling shutter effect of DSLR video slow sensor readouts, but their effective speeds are much much slower. I am not sure how much the improvement is, if anyone else knows.
A flash works by emitting one quick burst of light. If you manage to shoot a shutter speed faster than the sync speed, part of the image will likely be unexposed to the flash.
Some flashes have HSS, which emit many pulses of light. There are some disadvantages vs a higher sync speed, like inability to stop motion using flash (on the extreme end, often you use HSS at fast enough shutters anyway) and lowered max flash power
Sync speed is the fastest shutter speed that the camera can physically swing the mirror up and open the shutter curtains in sync with the flash.
Selecting a sync speed over the max sync speed will capture the shutter curtains or mirror in the exposure as it fails to "get out of the way" in time for the flash exposure. It appears as a uniform dark area. Select a lower shutter speed below the shutter max sync and adjust aperture or flash power to compensate.