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 involves very low 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 — which makes it one of the highest-temperature superconductors.
In the video it appears that a puck of yttrium barium copper oxide cooled by liquid nitrogen 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.