A magnet will levitate above a superconductor (or a superconductor above a magnet) because of the Meissner effect. This effect, discovered in 1933, causes the magnetic flux to be expelled from a superconductor. A superconductor is an example of a diamagnetic material.
A permanent magnet, such as the one above, has magnetic field lines emerging from the north pole and looping around the outside to re-enter at the south pole on the other end. When the magnet is hovering over a superconducting material its magnetic field lines are repelled or pushed away from the superconducting material because that material is diamagnetic. This means that the field lines emerging from the north pole of the permanent magnet cannot penetrate the diamagnetic superconductor. The magnet is forced to rise above the superconductor to give the magnetic field lines space to return into the south pole. That lifting effect leads to magnetic levitation.