FX and DX lens Nikkor will all work on DX cameras (the lower end Nikon DSLR models may have issues with some of the earlier model AF lenses). Older 35mm film era Nikkor Lenses may not work on all bodies.
With the D7000 you have a DX sensor. This has a size of (W x H) 24mm x 16mm as apposed to the FX which is 36mm x 24mm. As you can see the FX sensor is 1.5x the DX sensor. As already pointed out this "crop factor", as it is often referred to, is often incorrectly used to imply a higher magnification. The reality is that the FX lens is built to a focused image falling on an FX sensor. When used on a DX sensor the sensor size means that the total focused image falls outside the size of the sensor leaving a smaller, cropped, portion of the image falling on the sensor. The first thought is "Hang on I'm loosing part of the image". But you don't as what you see in the viewfinder is what is captured. How can that be? It is because the cropped outer portion of the image falls outside of the mirror box of the camera and so is discarded. This is just like the circular parts of the image that the lenses sees that don't appear because the film or sensor of the camera is rectangular.
A DX lens is designed for the DX sensor, so it is optimised to create a circular image that will fall over a 24mm x 16mm sensor. Thus when shown an FX sensor it isn't producing a large enough image the point it appears to the sensor that will cover the entire sensor.
When people refer to an FX lenses being a different focal length on a DX camera what is really happening is that the image on the DX sensor has a better pixel density than the same cropped region of the FX sensor.
e.g The D3s (FX) and the D300s (DX) both take an image of slightly more than 12 megapixels. But the cropped (DX) region of the D3s is only 5.3 megapixels.
For a good reference on using the D7000 and on Nikon DSLR in general get hold of Thom Hogan's guide e-book: Complete Guide to the Nikon D7000
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