The shutter of an SLR camera consists of two metal curtains that zip across the front of the film or sensor. At fast shutter speed the curtains appear to open and close instantaneously, but at faster shutter speed the second curtain trails the first curtain closely behind. hence, at any instant only a portion of the film or sensor is exposed.
Since a burst of flash is almost instantaneously, it has to fire at a shutter speed that slow enough for both curtains to fully expose the film or sensor. This is called the flash sync speed. Modern cameras will automatically limit the shutter faster to its flash sync speed.
Ironically, point-and-shoot cameras can shoot at high sync speeds because they use electronic shutters. Some newer SLR's that use electronic shutters in addition to the mechanical ones can also do the same.
Some cameras have FP (focal plane) mode whereby the flash will stay on as long as the shutter is opened. This enables shooting at any shutter speed. The effectiveness of the FP mode needs to be checked out.
Slow Sync mode: Remember this as "party mode". Shutter speed autonmatically slows down to capture background lighting under low light. Subject is sharp while background may be blurred.
Rear Curtain Sync: Flash fires just before the close of the second curtain. Used to freeze motion at the end of the exposure. E.g. motorcyle with trailing light.