As explained on this page, Earnshaw's theorem says it's impossible to have perfectly stable magnetic levitation where none of the fields are changing with time. But as is also discussed there, it is possible to have levitation that appears stable to the naked eye if the the currents that create the magnetic field continually adjust to small movements of the levitated magnet in such a way as to quickly damp out these movements. This can be done using artificial feedback with electromagnets at different positions that can adjust their fields in response to updates from sensors about the motion of the levitating magnet, as mentioned in the "feedback" section of that page, but there are also some examples of systems where the currents in the material naturally adjust in this way. One of these is diamagnetism, where the currents are just motions of electrons around atomic nuclei, and the electrons adjust to changes in the external field in such a way as to always create a magnetic field that's aligned opposite to the external field, and thus repels the magnet creating that field. A nice conceptual explanation of Earnshaw's theorem and how it doesn't rule out stable-appearing levitation with diamagnets can be found here. For most diamagnetic materials the response is very weak, but superconductors are an exception with a very strong diamagnetic response, which is why they are often used in dramatic examples of magnetic levitation like the superconducting hoverboard shown here, or the example of levitation with flux pinning shown in this video.
The Hendo hoverboard doesn't use superconductors or ordinary diamagnetism where the magnetic response is caused solely by the realigning of electrons in atoms though--instead it relies on the effect discussed in the "oscillating fields" section of the first page I linked to. In this case, a magnetic field that's designed to oscillate in the right way (which can apparently be achieved either using rotating permanent magnet or a varying current in a non-rotating electromagnet) will induce eddy currents in a nearby conductor, motions of large numbers of electrons that are not bound to particular atoms in the conductor. As mentioned on that page, and also in the oscillating electromagnetic fields section of the wikipedia article on magnetic levitation, this is really a large-scale version of diamagnetic levitation--apparently the swirling eddy currents adjust to continually repel the source of the oscillating field in the same way that individual electrons adjust in ordinary diamagnetic levitation. The wiki article links to this pdf article with a more technical discussion of how this works in the case of an oscillating field produced by varying current in an electromagnet, though apparently the Hendo "hover-engines" creates an oscillating field with rotating permanent magnets, as mentioned in the patent John Rennie pointed to in his answer. Here is a paper dealing with a numerical simulation of a rotating permanent magnet hovering over a conductive surface, although the configuration is a bit different (in the paper the axis of rotation is parallel to the surface like a wheel, whereas with the Hendo hoverboard the axis of rotation is perpendicular to the surface like a CD--apparently the type of rotation in the paper leads to forward thrust as well as vertical lift, which might be useful in future applications of this technology that need to be able to accelerate or brake).
I believe the "magnetic field architecture" refers to the way the oscillating magnets are designed in such a way as to produce an especially strong field below the board but a much weaker one above, using some type of Halbach array. This article mentions that the patent filed for Hendo's magnetic levitation system specifically referred to the use of a Halbach array.
As for "self-propelling", I don't know the details but this article may suggest it has something to do with a built-in system where pressure-sensitive pads are used to adjust the magnetic fields created by the different hover-engines, creating a bias in the strength of these fields which in turn should introduce a directional bias in the eddy currents pushing it from below:
Riding the contraption was a lot fun, but also quite the challenge:
The Hendo hoverboard doesn't ride at all like McFly's flying
skateboard. In fact, without a propulsion system, it tends to drift
aimlessly. Arx Pax founder and Hendo inventor Greg Henderson says it's
something the company is working on. "We can impart a bias," he tells
me, pointing out pressure-sensitive pads on the hoverboard's deck that
manipulate the engines. "We can turn on or off different axes of
movement." Sure enough, leaning on one side of the board convinces it
to rotate and drift in the desired direction.
Although on their kickstarter page they seem to contrast the hoverboard being "primarily ... self-propelled" with the idea of moving it by varying the magnetic field strength:
While our hoverboard is primarily intended to be self-propelled, the
actions which stabilize it can also be used to drive it forward by
altering the projected force on the surface below.
...so that suggests that they may be using the term "self-propelled" in the same way a skateboard could be said to be "self propelled", i.e. you gain speed by pushing against the ground with your foot.
