Taking it from the top (this is hard without being able to draw diagrams!)...
1) Changing a magnetic field passing through a conductor will create a current in the conductor. This is (basically) how a standard dynamo works; almost every form of electrical production relies on this method.
2) A current in a conductor will create a magnetic field. This is how an electromagnet works - the conductor is coiled around something so that the magnetic field produced is similar in shape and function to that of a bar magnet.
3) 1) and 2) are linked. Changing a magnetic field through a conductor will create a current (called an 'eddy current') in the conductor,
which will in turn produce another magnetic field. The direction of this 'created' magnetic field will always oppose the change in the magnetic field which created it in the first place (as described by Lenz's Law, if you want to look it up).
If that sounds too abstract for you, have an example. In
this video, a magnet is dropped down a copper pipe - and
something slows the magnet's fall. Let's imagine freezing time as the magnet is halfway down the tube - say that the north end of the magnet is upmost. At this point, the magnet will be creating two currents, whizzing in opposite directions around the copper tube (one below the magnet, where the magnetic field is increasing as the magnet gets closer to it, and an opposite one above the magnet, where the magnetic field is decreasing as the magnet gets further away). These currents will in turn be making the copper tube act as a two different magnets; and Lenz's Laws says that the created magnet above the real magnet will have it's North end uppermost (attracting the North end of the real magnet with it's closer South end), whereas the one below will have it's South end uppermost (repelling the South end of the real magnet with it's closer South end). These two created magnets will always act to slow the real magnet down, no matter which way up it is or which direction it's moving in.
4) Because most conductors have some resistance, the created current is never as strong as it should be, so the most that can happen is 'magnetic braking' - as shown in the youtube video linked to earlier in this post, or
here - this video shows that magnets (the solenoids, which are powerful electromagnets) can also slow down a conductor, as well as the other way around. The seond pendulum doesn't have anywhere near as drastic an effect because the gaps in it prevent any large currents from forming - proof that it's the currents that are important, not just a strong magnet.
5) However, a superconductor has
zero resistance. As such, it will 'lock' into a magnetic field - any attemps (below a certain threshold) to move it out of the magnetic field will cause it to become a magnet that opposes the movement. As such, you can move it into any position (the force we can exert is far more than the threshold) and it will stay there (the force due to gravity is less than the threshold). A simpler demonstration of that (with less powerful magnets) can be seen
here. Most superconductors only have this property at very cold temperatures - that's why both your video and this one show the superconductor as being really very chilled.
Further reading
Eddy current - Wikipedia, the free encyclopedia
Hope that's helped
EDIT: Tasha's link provided a nice extra 'quantum' twist to the whole thing (I thought that the 'quantum' monker was a bit of a stretch...), which explains why the superconductor doesn't just move sideways, without moving through the magnetic field at all. The reason it levitates is what I've posted, the reason it doesn't move sideways is
Flux Pinning.
