Is the Sigma AF 70-300 f/4.0-5.6 DG APO Macro compatible with the Nikon D5100 camera?
Maybe. This lens is made for Canon, Pentax, Sigma, Sony/Minolta, and Nikon mounts. You would need the Nikon version — F-mount. However, do also note that it won't autofocus on your camera, as the D5100 does not have the focus motor necessary to drive it.