A symmetry? is (in most classical situations) invariance under a group action.
A supersymmetry is invariance under a supergroup action.
Supersymmetry is particularly an issue in theoretical physics, which knows many models that suggest that the familiar symmetry of various action functionals should be enhanced to a supersymmetry in order to more properly describe fundamental physics.
This is doubtful and speculative. There are many models which have supersymmetry which is useful in their theoretical analysis, but the same models can be treated in formalisms not knowing about supersymmetry. Wheather the fundamental physics needs a model which has nontrivial supersymmetry is a speculative statement, and I disagree with equating theoretical physics with one direction in “fundamental physics”. I do not understand how can a model suggest supersymmetry; it is rather experimental evidence or problems with nonsupersymmetric models. Also one should distinguish the supersymmetry at the level of Lagrangean and the supersymmetry which holds only for each solution of the equation of motion.
The theory of supergravity is, as a classical field theory, an action functional on functions on a supermanifold which is invariant under the super-diffeomorphism group of .
Zoran: action functional is on paths, even paths in infinitedimensional space, but not on point-functions.