The Sigma 70-300mm F4-5.6 DG APO Macro is a lens that is available for both Sigma cameras and as a third party lens for many other brands, among them Nikon. These different manufacturers uses different lens mounts and this particular lens is available for Canon EF, Nikon F (FX), Pentax KAF, Sigma SA Bayonet and Sony/Minolta Alpha mounts. These mounts are not of the same physical size and not compatible with each other.
The lens that you were given is not ofnthe Nikon F type so it wont fit. The mount of your lens is called Canon EF. That mount is a bit bigger and looks like this:
Unfortunately you can't easily use this mount without using an adapter that will both remove autofocus, aperture control and more as well as losing image quality or not being able to focus at infinity. For a lens this cheap it's simply not worth it.