All particles that physicists have so far discovered - the building blocks of matter and energy in the universe - carry a quantum mechanical property called "spin". The nomenclature is purposefully chosen to suggest that this property is akin to imagining tiny particles spinning around. However, this is not really a genuine picture since particles in quantum mechanics are fuzzes of probability - instead of tiny spinning billiard balls. Yet, this spin property of particles does behave very much like the good old notion of spin that we know and love - except for one important peculiarity: quantum mechanical particle spin is quantized… This means that, when measured, its value comes out as a discrete multiple of a universal number: spin 1, spin 2, spin 3, etcetera times the universal number. But it's more interesting than that. We also find particles with spins that are half-integer multiples of the same universal value: spin 1/2, spin 3/2, spin 5/2, etcetera. And nothing else. So, spin values arrange themselves on discrete levels - like the steps of a ladder - and come in two categories: integer spins and half-integer spins.
These two classes have very different properties. Integer spin particles - called bosons - can be compressed together without resistance when there are no forces acting between them. Eventually, at high densities, they form an intriguing new form of matter called a Bose condensate. Superconductivity and superfluidity involve cooling things down to a point where such Bose condensates emerge - along with very peculiar macroscopic properties (see posts on superconductivity and superfluidity). Half-integer spin particles - called fermions - resist such compression: no two fermions can occupy the exact same state (this is known as the Pauli exclusion principle). The accompanying video shows a simulation of two fermions in a box. Quantum mechanically, the particles are simply probability lumps: red for high probability, blue for low. As the fermions are pushed together, they repel quantum mechanically; that is, there is no actual force law acting between these two particles! The repulsion is statistical in Nature. Notice in particular the interesting fringing pattern upon collision: remember these are supposed to be particles… that's quantum mechanics craziness for you (see post on visualizing quantum mechanics).
All visible matter in the universe happens to be fermonic: for example, electrons, proton, and neutrons are all spin 1/2 particles. All forces of Nature are transmitted through the mediation of force particles: and all such force particles are for some reason bosonic. For example, the electromagnetic force is transmitted through a spin 1 particle known as the photon. These two classes of lego pieces of the subatomic world then have very different roles and character. A few decades ago, some theoretical physicists started contemplating the possibility that fermions and bosons may in fact be related; that there is a deep symmetry in the natural laws that connects these what otherwise are very different types of particles. For a lack of a better term, we call it supersymmetry. There is no experimental evidence for this symmetry yet. However, interesting things happen when such a symmetry is hypothetically considered. For one, one realizes that it is possible to talk about all the known forces of Nature as a single force law (see post on Grand Unification)… very elegant and perhaps very true. Another consequence has to do with understanding gravity within the crazy quantum world: string theory, the only theoretical framework that purports to combine gravitational and quantum physics - requires Supersymmetry to be logically self-consistent. While supersymmetry does not require string theory, string theory does need supersymmetry.
An immediate consequence of the existence of supersymmetry in Nature is the necessity of a plethora of additional building block particles of fundamental physics yet to be discovered. These particles are needed so as to be paired with the ones we already know in a manner that makes the catalogue of fundamental particles more symmetric - supersymmetric. The new particle physics accelerator in operation in Switzerland, the Large Hadron Collider, may be able to see some of these additional particles - and hence confirm the existence of Supersymmetry in Nature (see post 1 and post 2 on the LHC). The discovery of Supersymmetry in the next few years would undoubtedly be the greatest scientific discovery of the new century. But then, the century is rather young.