Nikon makes two categories of lenses for DSLRs - FX format and DX format.
The lens format is tied to the sensor format in the camera body. The FX sensor is full-frame, meaning the sensor itself is physically the same size as a frame of 35mm film (36mm wide x 23.9mm high). The FX sensor is built into the D3, D3X, and D700. The DX sensor is smaller in size (23.6mm wide x 15.8mm high), and comes in the smaller DSLR's: D300, D90, D80, D70, D60, D40.
The image coming through the lens must fit precisely onto the sensor, so that what you see in the viewfinder directly translates to what you see in the photo. Thus, the FX lenses are designed to project a larger image onto the larger FX sensor, while the DX lenses project a smaller image onto the smaller DX sensor. The glass is different so that the image you see in the viewfinder is configured larger or smaller when it hits the camera body. Probably the easiest way to tell an FX lens from a DX lens (aside from the fact it's labeled on the lens) is to turn the lens around and look at the rear element of the lens; the FX has larger glass in the rear, while the DX has smaller glass in the rear.
Nikon does offer flexibility, however. If you use an FX camera such as the D3, D3X, or D700 you can still use both FX and DX lenses. If you use an FX lens, you get the full-frame image at the full resolution the camera offers. If you use a DX lens, the camera will automatically crop the image to account for the smaller image being projected onto the sensor. You also get a reduction in resolution. This is because the image being projected onto the sensor from the lens, is actually smaller than the sensor itself. Basically the camera adjusts for the fact that the outer edges of the sensor are not receiving any light/image from the lens. Net takeaway - if you have an FX camera, you can use both types, but you lose resolution with a DX. So it makes more sense to use an FX lens.
On the other hand, if you're using one of the DX format cameras listed above, you can also use both FX and DX lenses, but with different results. If you use a DX lens, the image will be normal - normal size and normal resolution. If you use an FX lens, you still get the image you see in your viewfinder, but the focal length of the lens is magnified by 1.5x. So if you set the lens to 100mm, your image is actually 150mm focal distance. You will still get the exact image you see in the viewfinder, because the magnification happens through the lens, and thus through the viewfinder. This actually is a good thing, if you're hoping to extend your range for wildlife or sports. But it can be a bad thing, if you're trying to shoot wide-angle. For example, an 18mm focal length would effectively become 27mm because of the magnification factor. Bottom line - for a DX camera, you can use either format at full resolution. An FX lens helps with longer shots due to the magnification, but can hurt when it comes to wide-angle.
Perhaps the best comparison/explanation I've ever seen is embedded below. Big props to Lilkiwiguy87 for taking the time and trouble to put this together. While I can't necessarily vouch for the musical selection, I must say the video was very helpful as I was researching between the D300 vs. D700, and accompanying lenses.
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