The best way I know to explain this is to simply illustrate it. So, I just took out my 12-24DX lens and my 14-24 non-DX lens and shot with both of them on the D700 at 24m.
Interestingly, the file size of the shot with the DX lens is 5.3mb, where the one shot with the non-dx lens is 11.6mb. This, of course, is because the camera crops the photo to the size of a DX sensor, which is the size that the DX lens is projecting onto the sensor. When you look through the viewfinder with the DX lens you see a box illustrated within the frame and this is the size of the actual photo that will be recorded.