The terminology was introduced by Serge Lang to enunciate a range of conjectures linking the geometry of varieties to their Diophantine properties.
The special set for a projective variety V is the Zariski closure of the union of the images of all non-trivial maps from algebraic groups into V. Lang conjectured that the complement of the special set is Mordellic.
A variety is algebraically hyperbolic if the special set is empty.
Lang conjectured that a variety X is Mordellic if and only if X is algebraically hyperbolic and that this is in turn equivalent to X being pseudo-canonical.
For a complex algebraic variety X we similarly define the analytic special or exceptional set as the Zariski closure of the union of images of non-trivial holomorphic maps from C to X. Brody's definition of a hyperbolic variety is that there are no such maps.