I just found this video on Youtube, with the description:
A thin superconductor layer (~1µm thick) is coated on a sapphire wafer. Quantum physics tells us that the magnetic field penetrates into the superconductor in the form of discrete flux tubes. The superconductor strongly pins these tubes, causing it to float in midair.
It looks real, but people can do some pretty impressive magic tricks (especially if they control the camera and can post-process the footage). Unfortunately, I don't know enough about quantum physics to know if this is Star Trek talk or real science.