Is it a crop? 
OK, you have a camera that has a 1.6 focal length conversion factor. Does that mean the camera crops the image? Does the camera magnify the image? Does it change the focal length of the lens? Something else going on?

Well, right off the bat we can eliminate the magnification or focal length questions. The focal length of the lens is a property of the lens and is not changed by what camera body it is attached to. A 50mm lens is a 50mm lens. No special voodoo magic in a camera body is going to change the focal length of the lens. What does change is the field of view of a lens. The lens projects an image circle. Different cameras make use of different angles of view within the projected image. Smaller sensor cameras discard the outer range of the circle. Is that a crop? You have to decide for yourself. I don't think it really matters. But some folks get very religious about this. Look at the figure here and decide for yourself if you want to say the camera is cropping out the outer range of the image circle or if you want to describe it some other way...

In the above example, the full rectagle is what your eye sees. The larger black circle represents the image circle projected by the lens. The blue rectangle represents what a full-frame 35mm size image sensor records. The red rectangle represents what a 1.6 crop-factor camera sees. You decide for yourself if you want to call it a crop of the full data or something else. Me? I'll just go out and make more photographs and not worry about it.

Why does my full-frame camera have a slower flash syncronization speed than my 1.6x crop camera?

Because there is more image area that needs to be fully exposed by the shutter on the larger image sensor.

Your camera's shutter determines how much time light will be allowed to reach your sensor. This time is measured in seconds and fractions of seconds. 1/30, 1/250, 1/500 are example shutter speeds. Most modern cameras use what is called a split-curtain shutter. This works by moving two curains across the image sensor. You press the shutter button and the first curtain starts moving to uncover the sensor and allow light to reach it. After a specific interval (which varies by shutter speed) a second curtain starts closing over the sensor to stop light from reaching it. In continuous light situations (the sun, an incandescent lamp, etc.) the entire sensor doesn't have to be uncovered all at once. You can have a "slit" between the first and second curtain travel across the sensor exposing a small area at a (very rapid) time.

However, when using flash this doesn't work. If the flash goes off while one of the curtains is in front of the sensor you get a black band along one side of the image. The flash synch speed of the camera is the shutter speed at which the closing of the second shutter curtain happens after the first curtain is completely out of the way. In the "old days" this was typically at 1/60th of a second (or slower). At 1/60 second the first shutter curtain opens, the sensor (or, in this case, film) is completey exposed, the flash fires, and then the second curtain starts closing. This way the flash was able to reach the entire frame of film. If the shutter speed was increased to 1/125 of a second the first curtain would open all the way, the second curtain would start to close, and then the flash would fire, causing part of the frame to not get any light from the flash.

In early 35mm cameras the curtains travelled horizontally. This meant that they had the full length of the film frame to open for the flash synch. As technology advanced vertical travel shutters became available and the snych speed was able to be raised to 1/125th and even 1/250th of a second for the full frame.

You might be realizing now that with a smaller frame (crop sensor) you can use higher shutter speeds because there is less frame area to be exposed. So the second curtain can start travelling before the flash fires and it will not cover the actual sensor area because the sensore is smaller. Another illustration...

This is a gross simplification of the situation, but should serve as a good example. Here we see the shutter curtains represented by the coral color area. If a flash was fired while the curtains were in this position on a full-frame camera part of the image would be blocked by a shutter curtain resulting in a black band across one side of the image. But at that same shutter speed the entire smaller sensor area would be exposed and no black band would appear. Now you get the idea of why a crop camera might be able to synch at 1/250 or even 1/500 of a second, but the full frame camera might be limted to 1/1/25 of a second.

Some may ask, "what about Canon's High Speed Synch?" When your Canon flash is in high speed synchronization mode you can take flash photographs at higher shutter speeds and not have the black band across the image. This is possible because in that mode the flash switches to lower power and fires multiple times (stroboscopic) in synch with the moving shutter slit. Instead of one big flash you get a quick series of flashes (too fast for most people's eyes to distinguish). I mention the Canon system here because that is the one I'm most familiar with. Other camera brand flashes may have similar capabilities. This only works with dedicated automatic flash systems. External studio flash systems do not have the electronic integration with the camera to be able to do high-speed synchronization.

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