Flash sync speed is the maximum shutter speed possible when using a flash. For most flashes, the flash sync speed, sometimes also referred to X-Sync speed due to the use of Xenon in the flash bulb itself, is around 1/200th to 1/250th of a second. When using flash, your maximum shutter speed is limited to the flash sync speed. This can be a limiting factor, particularly for action photography, as 1/200th or 1/250th of a second is often not enough to stop the kind of action being photographed.
Some higher-end camera gear is capable of higher flash sync speeds. Some models support up to 1/500th of a second, which is better for photographing action. There are also alternative flash sync modes for better camera gear and flash gear. Normally, flash is synced with the "forward" shutter edge, and fires when the forward shutter curtain edge has opened and is moving. An alternative sync mode that syncs flash with the "back" shutter edge. When shooting with back curtain flash, you can produce action ghosting, and freeze your subject at the very end of the exposure, which is sometimes a desirable effect for sports photography. Finally, there is "high speed sync." With this alternative sync mode, cameras may sync to flash at any shutter speed, even up to 1/8000th of a second on top-end models.
High speed sync does have some limitations. With normal flash sync, there is a single pulse of the flash. In high speed sync mode, the flash pulses continuously thousands of times a second. This ensures that the scene is illuminated for the duration that the shutter is open and accommodates the behavior of a camera shutter at such speeds. The drawback here is that to provide enough power for continuous flash pulses, the power of each flash is less, by around 1 stop per stop of higher shutter speed. Additionally, since the scene is illuminated continuously for the duration the shutter is open, the flash itself is not as useful for "stopping" action. This is often not that big of an issue, however, as the higher shutter speed itself is capable of freezing action (particularly at 1/4000th or higher.)
If you don't need high-speed flash sync, any flash supporting a standard sync speed will suffice. If you need to sync flash at extremely high shutter speeds, then you will need both a camera body and a flash that supports high speed sync. You won't be quite as limited with high speed sync, but keep in mind that the power of your flash will be a little less than normal. (Generally, this is not a problem at all, and you can usually open your aperture to compensate...but it is a factor to be aware of.)