I'll field this one. If you've ever tried to press two negative or two positive poles of a magnet together you should know that they oppose each other in the most sporadic of ways. In other words, they resist contact and also tend to fly off from each other. Mag Lev simply uses this property in such a way that the repulsion effect is obtained without the problem of the two magnets to fly away. For example, in Mag Lev, the magnetic rails simply won't budge because they're nailed down. And the train itself won't budge because the gravitational force properly attains equilibrium with the sporadic tendency for the two magnets to fly away. No cooling effects needed in this case. Only properly balanced weights and magnetic strengths with well designed mechanical constraints.
The other effect is different. It's a property of the superconductor. I read all this on Wiki, so I don't know what I'm talking about. But I can take a stab at it.
The thing you see hovering above the magnet is a superconductor. The word conductor simply means that the substance conducts electricity. In other words, it allows electrons to move freely about the substance. This property is most often found in metallic substances. Other non-metallic substances are often not conductors in the sense that their electrons stay put, often because they are integral to the stability of the substance.
A superconductor is a substance that allows electrons to move freely about without any form of resistance what so ever. In the case of a normal conductor the electrons meet with small forms of resistance. But not so with the superconductor. The superconductor is brought to a very low temperature. When it approaches the state where the electrical conductivity of the substance is nearly absolution, the magnetic field within the substance is diametrically expelled.
Normally, a magnetic substance forms a field of magnetism that emanates from the very core of its existence, locking and trapping other magnetic things in a very horrible way, I'm sure. But when it gets very, very cold magnetism decides it's had enough, says, "righto. We've had enough. If you're not going to come out of the cold, we're leaving!" And just like that it's gone and the electrons rejoice for their magnetic over lord of clinginess has given up total power.
But then this very cold object is put inside the context of another magnetic field, like the object above or below which the superconductor in the video is "pinned". Here the magnetic field of the other object attracts the superconductor and holds but not as it would were the substance to still be in the bond of its own magnetic field. So this is why the superconductor is trapped above or below the other object.
Another brilliant form of levitation can be seen in the night sky. The moon levitates above the earth in a similar fashion according to the mutual bond of gravity. How exactly these objects stay in relative position with one another is still a topic of discussion among scientists and lay men alike. However, one thing is clear. The moon and the earth are like atoms. Anyone can see that though gravity binds them, there is truly no contact betwixt the two. Like the atoms in our being, no contact exists, though consciousness binds us to the material fabric.