https://en.wikipedia.org/wiki/Diamagnetism
https://www.kjmagnetics.com/blog.asp?p=diamagnetic-levitatio...
...and superconductors are usually perfectly diamagnetic.
* ferromagnetic - attracted to one pole of a magnet but not the other (in a given orientation), this is what everybody thinks of when they think of "magnets"
* paramagnetic - attracted to both poles, i.e. stuff that sticks to magnets
* diamagnetic - repelled by both poles, except in superconductors, this effect is very weak compared to the forces experienced involving ferro-ferro or fero-paramagnetic materials.
There isn't another category, everything fits in to one of those buckets.
Saying
>Just so everyone is on the same page, static passive diamagnetic levitation is possible with materials like pyrolytic graphite.
is a bit deceptive, as what people know as "magnetic" materials are ferromagnetic.
https://www.sciencedirect.com/science/article/abs/pii/S03048...
And many others besides. Halbach arrays are fascinating.
Halbach arrays with compensating coils have been proposed for some interesting applications, such as low loss flywheels for electrical storage. I don't know if that ever got commercialized but I do recall that some prototypes were made by a US company. I can't find a reference to it though.
As far as I know a stable arrangement of permanent magnets levitating is impossible without a baring surface to keep them aligned. (i.e. free floating levitation is not possible without active control)