This levitation effect is explained by the Meissner effect, which describes how, when a material makes the transition from its normal to its superconducting state, it actively excludes magnetic fields from its interior, leaving only a thin layer on its surface.
When a material is in its superconducting state (which occurs at extremely cold temperatures) it is strongly diamagnetic. This means that when a magnetic field is externally applied, it will create an equally opposing magnetic field, 'locking' it in place.
A material called yttrium barium copper oxide can be turned into a superconductor by exposure to liquid nitrogen, as one of the warmest superconductors.
For the video, it looks like they just made a clump of yttrium barium copper oxide and cooled it with liquid nitrogen in the shape of a hockey puck. This 'puck' is repelling the magnets embedded on the handheld device. It also shows that the angle of the magnet can be locked in a magnetic field. Later in the video the puck can be seen to zoom round a circular track of magnets, in the same way that Maglev high-speed trains do.
Virtual particles ftw!