I was very confused also initially when I started exploring DSLRs, but I got the picture after going through a few websites.
Field of View (FOV) is basically how wide and how long your photo covers, which is about the camera's sensor size. APSC DSLRs have smaller sesnor sizes compared to the size of a 35mm film and thus can only take a photo that has the edges cut off. Imagine shining a torchlight from a distance onto a small piece of paper in the dark, only a small amount of light will fall on the paper. The rest of the light gets to the outside of the paper. Now point the torchlight to a larger piece of paper from the same distance. Don't the larger piece get more of the light than the smaller piece? Another example, if you cover the sides of you eyes with your hands what do you see? Now remove your hands, you will have a wider view. Explanation sounds rather crude, but hope it is clear
So basically the crop factor (1.5x for Nikon) is the size of the photo the sensor takes relative to the dimension of the photo 35mm film takes, just like A3 size is 2 times (2x) of A4 size. Like what diediealsomustdive said, the physics (the properties of the glass that creates a sharp image at a certain focal length) of a DX lens and FX lens is the same, the difference is that the DX lens produce an image that has the edges of the photo cut off from the FX lens (the torchlight example). You can still use a FX lens on a DX camera, just that the edges of the image formed at the back of the lens will not be captured by the sensor. You can still use the DX lens on a FX body, misconception that it can't be used. The problem is
whether you want to use it, cos they will only use the centre portion of your FX sensor, that is you will get a cropped photo instead of a full 35mm FOV. Of what I remember the newer FX bodies have the function of switching to use 1.5x crop. You can read more about it here:
http://www.kenrockwell.com/nikon/dx-lenses.htm