Superconductivity is an affect where metals, cooled below a certain critical temperature no longer exert any resistance on a current passing through it as shown in the graph (provided by openstudy.com).
The formation of this affect lies with quantum physics and specifically Pauli’s Exclusion principle. This principle is that no two fermions have the same quantum number (in essence they can’t exist with the same energy or in the same place). Electrons are fermions, both the ones flowing to produce the current and those bound in the atoms of the conductor. This and the local contraction of the material around electrons causes resistance. Now when the material is cooled down an electron can form a bond with another and a species called a Cooper Pair is formed. These pairs or electrons don’t act like fermions, being able exist homogeneously with all other Cooper Pairs in a Bose Einstein condensate and so resistance caused by Pauli’s principle doesn’t apply.
Superconductors also demonstrate something called the Meissner effect. When a magnetic field passes into the superconductor a small circular current is created which in turn produces a magnetic field in the opposite direction. Since this current never decays all magnetic fields get expelled from the material and are forced around it instead. This creates a very stable hovering above the magnet (this time provided by www.extremetech.com).
Superconductors are often used either for this levitating ability or for the extremely powerful electromagnets they can be used to create. They can be used in maglev trains, magnetic resonance imaging, electromagnetic confinement (in high temperature physics) and particle accelerators.