It's a combination of camera and strobe capabilities. The camera shutter determines the speed at which it can execute a normal flash sync - the first curtain exposes the sensor, then the flash fires, then the second curtain moves in and closes the sensor. The shutter curtains don't move instantaneously though, and the speed at which they move out and in determines the overall shortest possible exposure - while they're moving, the sensor is still getting some light. In order to provide a shorter exposure, the second curtain starts closing before the front one is fully open, creating a strip of light that races across the image, and this precludes using a normal flash, because if it fires when only part of the sensor is exposed then you get underexposed areas where the curtains were covering the sensor at the moment that the flash fired. Some cameras have shutter curtains that move exceptionally fast, like the Nikon D50 that allows 1/500s flash sync; some are much slower, like Nikon 1 series that are generally limited to 1/60s flash sync speed. Most Sony interchangeable-lens cameras are limited to 1/160s, or, in some limited circumstances as Pavel mentions, 1/200s.
Here is an animation showcasing this process: https://cdn.fstoppers.com/styles/full/s3/media/2015/04/fstoppers-high_speed_sync-_shutter-_operation.gif
Fixed lens cameras usually have diaphragm or 'leaf' shutters located in the lens - these have a much smaller distance to cover when opening and closing, and thus allow for much faster flash sync speeds. For example, Sony RX100 series cameras can sync flash at up to 1/2000s. Note, however, that full-power flash dumps usually take around 1/320s (for commonly used underwater strobes such as Inon Z-330/S&S YS-D2; figures may differ for land strobes), so syncing your flash faster than that cuts into strobe power at the same rate that it cuts into ambient light power.
The way this limitation is overcome for interchangeable-lens cameras with curtain shutter is so-called high-speed sync where, instead of emitting a single powerful pulse, the strobe emits a series of smaller flashes, typically modulated at 40kHz while that thin strip of exposure races across the sensor. Alas, this has the unfortunate effect of massively reducing effective flash power, and in underwater strobes, support for this mode is almost nonexistent. The long since discontinued Olympus UFL-2 was effectively a land strobe (Olympus FL-36) in an underwater housing, and thus inherited its ability to do HSS with select Olympus cameras using their RC triggering. The upcoming Retra Prime and Retra Pro flashes claim HSS capability with triggers that can drive this mode using LEDs, but they are not shipping yet, and neither are compatible triggers. Once they're out (C'mon Oskar!) we'll know more about how useful this mode is in practice. As mentioned above by Pavel, Sea & Sea YS-250 Pros with their long burn time at full power allow for a sort of of pseudo-HSS, but these are also discontinued. With other strobes, including YS-D2s ,you can't sync any faster than allowed by your camera's shutter mechanism.