What happens is that full frame sensor cameras have the lens mounted at the same distance from the sensor as film in a 35mm camera would be. So your focal length in both types of camera would yield the same result.
In the APS-C cameras, the sensor is closer to the lens. What this does is cause the focal length to be multiplied by a certain amount, with the picture from the lens being cropped. In essence, these cameras give you an extra bit of zoom, so your 200mm lens focal length would be multiplied by around 1.3 giving you a lens with an apparent focal length of around 260mm.
Many factors can affect the appearance of the camera. You a more likely to get images that "pop" if you buy cameras further up the hierarchy, as well as buying lenses with all glass components instead of a mixture of bass and plastic.