First off, you need to design a stable magnetic track. This is actually simpler than you might think; you just need three magnetic “lines”, where the outer two go in one direction, and the inner one goes in the opposite direction:
Next, you need a high temperature superconductor, typically made of Yttrium, Barium, and Copper Oxide. This way, when you get this thing down to the temperature of liquid nitrogen (77 Kelvin), all of the resistance goes away. (That’s why, in the videos, the levitating thing is always smoking; that’s the fact that it’s only 77 Kelvin!)
When you bring a material like this close to — but not in contact with — a magnet, it magnetizes as well. Typically the little eddy currents (below) that magnetize this material decay after a short amount of time.
But when it’s superconductor, there is no magnetic field inside! That’s what the Meissner effect tells us (for those of you who followed the hint yesterday). In other words, the superconductor expels all the magnetic fields inside of itself, creating a perfect diamagnet, which simply means that it repels the permanent magnets below it.
And that’s why it levitates! But, how do you get it to always stay the same distance away from the permanent magnet, even when it’s upside down?!
Well, that requires one more thing, called flux pinning, which means you need imperfections in your superconductor. The effect of flux pinning keeps the magnet from floating away, and keeps the superconductor at a fixed height away from the magnet. The forces of the magnet and of the flux pinning are very large compared to gravity, and that’s why you can turn it at an angle or even upside-down!