Quantum theory and special relativity have been unified for a long while in the framework of quantum field theory, and supersymmetry alone does not take us any further in that direction.
In particle physics there are two types of particle. The first type are the fermions. These are the particles which make up matter, like the electron and the quarks that are the building blocks of the atomic nucleus. The second type are the bosons, which mediate the forces, like the photon which mediates the electromagnetic forces. The difference between them is a property called spin, which doesn't really have any classical analogue, but you can imagine that each type of particle spins on its axis at a certain rate. It turns out that this spin has to be quantised, so it can only take certain discrete values, which we can label by 0, 1/2, 1, 3/2, 2,... The fermions have the property that their spin is half-integral, so 1/2, 3/2 etc and the bosons have integer spins 0,1 etc. For example, the electron and quarks have spin 1/2 while the photon has spin 1. There are lots of consequences of this difference which means that fermions and bosons behave quite differently.
Supersymmetry is an extension to the ordinary symmetries of spacetime, like rotations which can turn 'North' into 'East', to include symmetries which turn bosons into fermions and vice versa. This means that every particle must have a 'supersymmetric partner'. While we haven't got any real physical evidence for it, it is very attractive theoretically for a number of technical reasons. In particular, it is an important part of string theory, which is where it first appeared historically, though it can exist without string theory.
It's possible that we might see the first hints of supersymmetry at the LHC quite soon (even before the elusive 'Higg's boson') so watch this space!
