It's a superconductor [a material which has no electric resistance (at a certain temperature)]. Notice that liquid nitrogen is poured into the reservoir. When certain metals and ceramics are cooled to temperatures close to absolute zero (-459 F or -273 C), they have no electrical resistance allowing electrons to flow through them freely - that's what's necessary to cause the levitation.
The floating effect results from the other object being a regular strength magnet and at room temperature. The superconductor is nonmagnetic when the nitrogen is added, the meissner effect occurs and the superconductor is now perfectly magnetic. The smaller, regular magnet now floats as if it was suspended over a large regular magnet (like an electromagnet).
Thanks to the commenters for the info - Scienott