I tried to add this as a comment, but it is too long so I am making this an answer instead. This is not my text, but the text of one of the commentators on the video:
"Superconductors are of two types, which are defined by their
Meissner effect. One type repels magnetic fields, which will levitate
the superconducting object. A type I superconductor becomes a perfect
diamagnetic material, which exhibits a magnetization in the opposite
direction of an applied magnetic field. The Meissner effect creates a
complete diamagnetic material so that no magnetic field lines are
present in that material. I doubt this will suspend the object
against gravity by putting it on bottom, for the magnetic fields in
opposition will impose a force on the superconductor in the same
direction as gravity.
There is what might be called an anti-Meissner effect where the
superconducting material collimates magnetic flux lines into narrow
tubes or vortex fluxes. If the magnetic field at large is not
perfectly uniform it takes work to move the object through the
magnetic field and so energetically it is favorable to remain in a
region with B_in and B_out remains the same. This is the
Landau-Ginsburg effect and is found in type II superconductors. I
think that this is a case of a type II superconductor."
This sounds right to me and explains what is meant by quantum locking since superconductivity is a macroscopic quantum phenomenon that is effectively locking the magnetic flux into specific tubes in the superconductor. The force that opposes gravity is, of course, magnetic so we are not talking about any kind of new force of nature.
When he uses his hand to move the superconductor, he is using enough force to make the magnetic flux tubes be rearranged but apparently the force of gravity is weak enough such that it cannot rearrange the flux tubes by itself. So I predict that if you added enough weight to the puck, it would fall :)