First, a disclaimer. This is a very complex and scientific subject, and I'm going to do my best to explain this topic in very simple terms because this is an oft-misunderstood subject. So if you're an advanced super-photographer, just feel free to tell me what I missed in the comments below, but know that I probably skipped it on purpose.
The path to understanding flash sync speeds begins with understanding the shutter in your DSLR camera. The shutter is made up of two curtains that quickly slide from the bottom of the frame to the top of the frame, thus revealing a slit of light. The two curtains move very quickly. In fact, on most cameras, if the shutter speed is under 1/200th of a second, then the entire sensor will be revealed before the second curtain begins to slide across the sensor.
Like shutters, the light coming from a flash is extremely fast. One burst of light from a flash begins and ends in 1/500th of a second or faster. The trouble with flash is that, if a shutter speed of over (or faster than) 1/200th of a second is used, the flash can only illuminate the portion of the image that is not covered by the sliding shutter. That leaves part of the image correctly exposed, and part of the image dark and untouched by flash. Woops! In case you are wondering what that looks like, check out the disastrous piece of photography to the left.
So what can you do to fix this short-coming of flash photography? Two answers.
First, if you are using the flash made by the manufacturer of your camera, you can often bump it up a little higher–often to 1/250th or 1/320th. There is a setting in your camera menu that will allow you to enable this function. This is convenient for shooting a wedding or other event where you'll often have the flash on the camera and bounce the light off walls or the ceiling.
Second, if you buy one of the new fancy flashes from either Canon or Nikon, you can use what is called high-speed sync. This turns your flash into a machine gun, but with less destruction. It makes the flash fire several low-power extremely quick bursts of light. The multiple bursts will occur at planned times during the exposure so that the whole frame will be illuminated. With high speed flash sync, photographers can use shutter speeds of over (or faster than) 1/1000th of a second. If you turn on high-speed flash sync and you only detect one burst of light, it is because the bursts occur so quickly and in such rapid a succession that it is impossible to detect the multiple bursts.
There you have it… flash sync speed. Have a question about something that wasn't quite clear? Leave a comment below!