the six realms of reincarnation

The "six realms" is acutally a strong reference to Buddhism. There are a ton of things about Mukuro that stand out with this particular belief - the six realms of his illusions, Mukuro's stronghold of power, ties hand in hand "The Six Realms of Existance". Remember, I'm only here to inform, not convert. It's basically a wheel, a cycle of rebirth. Not entirely going into complete detail about it, but in basic: the belief is that one is reborn through each realm depending on different virtues and the belief of karma.

These various realms each have a "skill" (I think the skill part is fictional) that Mukuro has acquired throughout his various rebirths. Instead of saying "realms", Mukuro describes them as "paths"..

First path - Realm of Hell: Described to cause endless nightmares; can create illusions.
Second path - Realm of Hungry ghosts: Can use enemies' abilities when they are possessed by user.
Third path - Realm of Animals: Can summon dangeous, real animals.
Fourth path - Realm of Asura (demigods): Battle skills sharpened.
Fifth path - Realm of Humans: Aura increases, strength increases.
Sixth path - Realm of Heaven: The ability to control other people.

Mukuro states that the Fifth path is currently the world that they live in, the world of humans. Out of the six, Mukuro refrains from using the fifth because it's dangerous, and the strongest of them all. As you can see, each path has it's own merit during battle, and Mukuro has strongly kept his word at present -- the fifth path, i've noticed, is th only one that isn't being used so often. It may be because of Tsuna's purification, who knows, but the six realms/paths are the core of Mukuro's usefulness to the Vongola.

I think that the Reborn! series in particular, the characters that have the ability to be an illusionist, have to acquire these skills as well.

For further indepth explanation of the "six realms", you can look to this page for symbolism information as I am not knowledgeable in this specific subject.