Actually, the Kings DO have one major thing in common. This one thing is what sparked their entire power struggle in the first place.
They eat humans.
Their power struggle comes in the form of "How should we handle this human-eating business?" Yomi wants to eat humans as he pleases, with no regard for secrecy или rationing. Mukuro just wants to eat, but not make undue trouble или be a giant pig about it. Raizen wants to leave the humans alone (falling in Любовь with Yusuke's ancestor was the impetus for this mindset).
At any rate, what makes a mazoku?
Well, looking at the way the word is written in Japanese, we have two kanji on our hands. The first is "ma," written to mean "devil," and the секунда is "zoku," written as "family." So if Ты want to take the literal translation of the word, we get "devil family" или "family of devils." This can be further simplified to mean a powerful being separate from humanity--AKA one of YYH's many demons. The word is an umbrella term to indicate any demon in existence, is my reckoning.
Throughout the (original Japanese) series, the characters from the Ningenkai/Human World and the Reikai/Spirit World use the word "youkai" to indicate what we refer to in English as "demons." Some demons in the series use "mazoku" instead of "youkai," so I think the terms are interchangeable synonyms and not different breeds или creatures.
I believe the only time "mazoku" was used in the English dub was when Yusuke's heritage came up--he had "The Atavism of the Mazoku," is what I remember. I think they started using that term in the English because "The Atavism of the Demon" sounded stupid and they needed a Japanese phrase.
At any rate, I think "mazoku" pretty much means "demon."
Although, and I feel I'm obligated to add the following: Ты might be able to extend the meaning of the word "mazoku" to "a particular demon that eats humans" if you're working with the English dub. People keep referring to Yusuke as a mazoku and Yusuke keeps making a big deal about how he won't eat humans, so the word and the human-eating seem connected (at least in the English). I'm not sure that that connection holds up in the Japanese, though. The English dub changed a few things, sometimes for the better clarity, sometimes for the weird.
The Mazoku are a special kind of demon. They are extremely powerful and passes on their power through many generations. They do this to satisfy their curiousity. And, in a rare case, happens when a demon and human have a child so the demon needs to pass on the power. The demon genes will only awake if the descendant is strong enough.