Before you can understand FP High Speed Sync, you have to understand what FP means and how the Normal Flash Sync works.
FP stands for Focal Plane and it refers to the type of shutter used in most modern DSLR cameras. A focal plane shutter is actually two precisely timed curtains positioned between the lens and the sensor that can either block light from hitting the sensor or allow light to hit the sensor. The reason there are two shutter curtains is to be able to get much higher effective shutter speeds.
It is important to understand is that these curtains open and close in exactly the same amount of time. So the the shutter speed is set by timing between the start of the first curtain opening and the start of the second curtain closing.
Notice that the entire sensor will be open to the light at every shutter speed up to the speed of the curtain movement itself. This is the maximum normal Flash Sync Speed. To say this another way; at all speeds up to the maximum normal Flash Sync Speed, the first curtain completely opens before the second curtain begins to close. At any shutter speed higher than this, the second curtain will begin closing before the first curtain gets fully open, thus never exposing the entire sensor at any one time. At really high shutter speeds, this results in very narrow 'slit' of light that travels across the sensor.
These two curtains travel vertically across the opening from the top to the bottom, and this in itself causes some strange effects when using high shutter speeds (small slit). If you shoot something that moves horizontally really fast like a race car from the side, you can sometimes see that the wheels seem to lean forward a bit because the top part of the wheel was exposed after the bottom part and the top moved forward a little bit as the slit moved from bottom to top. Of course, the image is inverted on the focal plane, exposing from the bottom to the top of the image, and that's why race car wheels lean forward even though the shutter that moves from top to bottom. Also, this effect only occurs if you don't pan with the car. You can totally eliminate the forward leaning effect by panning.
As I mentioned, a focal plane shutter mechanism moves the curtains at a very precise speed. This speed is determined during manufacturing of the mechanism and is governed mostly by how recently the shutter was designed. In older 35mm cameras, this speed was 1/60th second, but with time, shutters got faster and faster, and in the new D300 this speed is 1/320th sec. And those race car wheels lean much further forward with an older 1/60th shutter than with the D300 1/320th shutter.
Once you understand how the shutter works, you can begin to see what needs to be done to synchronize the flash. Depending on the design of the flash, the length of a full power flash will vary. In an SB800, the maximum flash lasts about 1/1050th sec.
The key point is that the flash, firing at 1/1050 sec is much faster than the curtains which move at 1/250th sec on the D200 (or 1/320th sec on the D300), so to synchronize the flash it must fire at some point after the first curtain has fully opened, but before the second curtain begins to close, so the sensor is completely exposed to the light from the flash coming through the lens. You may also see that you have a choice of when you fire the flash; either right after the shutter opens (front curtain sync) or right before the shutter closes (rear curtain sync). The timing of the flash has profound effects on the light trails when using very long shutter speeds. It will place the light trails either in front of the subject when using front curtain sync and behind the subject when using rear curtain sync. Rear curtain sync usually looks more natural for light trails.
Up to this point I have been only discussing the normal flash sync. However, Nikon has added a mode called 'Auto FP High Speed Sync'. You select this mode in the menu on the camera. In this mode, the flash duration is stretched so that it fires continuously for the full curtain travel time, ie, 1/250th sec on the D200. So, the the flash pulse that normally takes 1/1050th sec is now stretched across 1/250th sec. The official name for this operation uses the word 'Auto' in front of 'FP High Speed Sync', because in this mode the flash will sync normally and fire normally below the flash sync speed, but it switches automatically into High Speed Sync (stretched flash) when the shutter speed is set higher than Normal Flash Sync Speed.
This all sounds like a pancea until you find out that stretching the flash pulse and using higher shutter speeds causes the power of the flash to be reduced from what it normally is. In fact, the times where you would really like to be able to use a mode like this, like fast action sports in bright daylight, you usually can't, because the high speed flash sync mode is not powerful enough.
In fact, the power becomes less and less as you increase shutter speed, because the slit gets narrower and narrower. So, in this mode the flash is dependent on aperture and shutter speed, and if the shutter speed is increased to 1/8000th sec, the power gets so low that a subject would have to be less than four feet away.
However, when stopping action is not the goal, and a really high shutter speed is not required, then FP Sync can be very useful. In fact, Auto FP is excellent for shooting portraits in bright daylight. Then, you can use a shutter speed well above the normal sync of 1/250th coupled with a wide aperture to blur the background which greatly helps isolate the subject. I typically use camera A mode, ISO 100, FP Sync, f/ 2.8 and 1/1000th in bright shade, and I get a flash range of about 10-15 feet. If I am in really bright light, like on the beach, the shutter has to be around 1/1600th, and this reduces the range to about 8-10 feet which is still reasonable for portraits.
If a greater distance than 10 feet is needed in bright daylight, there is no choice except to use regular flash sync and accept the requisite smaller aperture. This happens frequently when shooting a party outdoors, where I typically shoot in camera S mode, regular flash sync, ISO 100, 1/250th, and f/7, and I get acceptable fill out to 20 to 30 feet or so.