(This video has no sound) Explanation: "The bottom piece is cooled down so it gets superconductive. The one on the top is a magnet.

The magnetic field from the magnet is causing the electrons to move in the superconductor (superconductive means electrons can flow with NO friction, so there is no resistance at all). The moving electrons in the superconductor generate a magnetic field much the same way as an electromagnet. The magnetic field generated exactly mirrors the magnet's own field, so it floats!!

More about this experiment: 

"High-temperature superconductors" are generally considered to be those that demonstrate superconductivity at or above the temperature of liquid nitrogen, or −196 °C (77 K).

Conventional superconductors, by contrast, require temperatures no higher than a few degrees above absolute zero (−273.15 °C or −459.67 °F). Though it is extremely cold by everyday standards, in the field of superconductivity, 77 K is considered "high temperature".

The most common "high temperature superconductor" that people use for these kinds of demonstrations is called Yttrium Barium Copper Oxide (YBCO). It's a kind of ceramic. 

It is being cooled by liquid nitrogen to −196 °C (77 K) in this experiment.

Definitions: "Superconductivity" is a phenomenon occurring in certain materials at extremely low temperatures (on the order of negative 200 degrees Celsius), characterized by exactly zero electrical resistance and the exclusion of the interior magnetic field (the Meissner effect)." 

The defining property of superconductors is the Meissner Effect, i.e. the very small penetrative distances of magnetic fields into the material.  

Note: He is not wearing a glove because he is not touching the
Yttrium Barium Copper Oxide (YBCO) ceramic which needs to be cooled. The magnet is not touched by the liquid nitrogen.

Further Links:
Make your Own Superconductor
Wikipedia:Meissner Effect