Supersymmetry is an extension of the properties of space-time symmetry using hypercomplex algebra (such as Clifford and Grassmann). In the same way that complex numbers are powerful tools in geometry and analysis, the algebra of complex numbers can be extended to cover notions of space-time in 4 dimensions (and more). The 'square root' of the vectors of space-time can then give rise to objects called spinors which, as their name indicates, are closely related to rotations in space-time. The result is an algebra, a geometry and a generalised analysis such that supersymmetry is sometimes called "the square root of geometry".
Supersymmetry was first discovered in string theory and studies aiming to generalise the theory of Lie groups and algebra around 1970, its applications in theoretical physics and especially in algebraic and differential geometry have proved extremely rich and powerful.
Just as there are translation and rotation generators, there are also supersymmetric transformation generators. Supersymmetry is initially global, it does not depend on its point of application, and in a single multiplet it associates integer spin bosons and half integer spin fermions. The main advantage of 'susy' is to have better control over the renormalisation problems inherent to relativistic quantum fields and to solve certain problems (it is hoped) in the grand unification theory (GUT). It predicts that all known particles can have a supersymmetric partner: the bosonic photon must have an associated fermionic photino and the electron (fermion) a selectron (boson).
However, if the symmetry were exact, the selectron should have the same mass as the electron and hence should be produced in large numbers in accelerators and in cosmic radiation. The symmetry must therefore be broken by a process analogous to that of Higgs.
These particles have been the subject of intensive experimental research for decades and it is hoped to observe them in the LHC.