What you Need
Superconductors are very bizarre materials at room temperature they behave like a normal metal or ceramic, electrically or in any other way, when they are cooled below their critical temperature(Tc) their resistance entirely disappears to zero.
Possibly the most beautiful and accessable result of this is magnetic levitation.
What is going on?
If you more a conductor towards a magnet you change the amount of magnetic flux in the conductor so you induce a circulating current in a conductor. This current produces a magnetic field which acts against the movement, just like in the magnet dropping down a pipe experiment. If the conductor has no resistance (its a superconductor) the current will never die away, so the superconductor will repel the magnet and levitate.
If the superconductor was perfect it would always be repelled by a magnet, however high temperature superconductors, which work at 77K (-196°C) or higher will allow magnetic flux into them if they are cooled in the flux or if you apply a large enough force. This flux gets pinned on slightly weaker parts of the superconductor. The superconductor is now an electromagnet, but it will still resist movements of the magnet, so now instead of just repelling, it is locked in one position over the magnet. This is why the superconductor didn't fall off when it was turned upside down.
Boeing is trying to use this effect to make superconducting bearings which in a vacuum will have far lower friction than a mechanical bearing, so they can build large flywheels to store excess electricity when the demand is low and return it when the demand is high, using a superconducting bearing the time the energy can be stored is measured in weeks not hours with a conventional bearing.
David Cardwell who demonstrated the levitation to us is also using this effect to produce large superconducting magnets using this effect which can be used for GP surgery MRI machines instead of the more expensive low temperature superconducting magnets which require liquid Helium to cool them.
Superconductivity have many other uses, as power cables, large magnets, sensors etc. find out more in "the history of superconductors" article
What to do
What may happen
Why does it happen?