Makai has become a very fascinating setting recently for anime and some games. Why? Because it appears to be owned by no one as a concept (though has been expanded on) but is way too cool to ignore. Thanks to Makai, the Aeternalae concept has finally found stability that it's not known for its entire lifespan prior. Makai and its Demonic residents can share a comparison to Tolkien Elves, not because generic Mazoku look like them, but the fact Tolkien can't claim ownership of Elves and now they're everywhere. While Mazoku, Makai, and Maou aren't always interlinked, two of the three are usually present if not all three. This thread is going to be used for a semi-glossary of information I know about the three terms so people will have a reference point for any Makai-related stories I submit.
Makai
(Good Translations: Evil World, The Dark World)
(Bad Translation: The Netherworld -- Makai has ghosts and Demons do enslave souls, but it isn't a realm for the dead)
The homeworld of the various races united only by the term "Mazoku", Makai lies parallel to the real world, except with twisted landscapes and even more twisted inhabitants. In Makai, power is law -- the strong live, the weak die. Various Demon factions have been known to fight each other in Makai to claim the right to the title of "Maou" -- absolute ruler. Even though the word Makai is absolute and isn't really meant to be pluralized, either only one anime/game showed Makai correctly or there are number of alternate worlds (and thus numerous Maou).
Mazoku
(Translations: Evil People, Dark Magic Race, Demons)
(Bad Translation: Mystics, courtesy of Squaresoft of America on two occasions though it did offer me the base for my "Sommystic" handle)
The blanket term for all of the Demons in Makai, or at least the more humanoid ones. Mazoku are sort of a race of their own nowadays, portrayed as having triangular or pointed ears like an elf (I'm unsure of where to point the blame for this trend, but I'm tempted to blame it on Magus and Flea -- Laharl and Zetta don't help either). However, they're still more of a similar-looking ethnic group than a race. Those who are part of the Mazoku tend to create relationship problems for others of their kind who are not (Such as in the case of the Aeternalae). The Mazoku harbor a strange animosity for humankind and look to invade human-owned worlds (Ningenkai) frequently, such as in Yuu Yuu Hakusho.
The singular form of "Mazoku" is "Akuma", however you won't hear the word "Akuma" as much due to both the Street Fighter character creating confusion and the fact that Japanese etiquette defines people in groups rather than individual samples.
Maou
(Translations: The Devil, Lord of all Evil, Demon King)
(Bad Translation: Overlord -- the word is neither sufficiently sinister or descriptive enough)
The most coveted rank among the Mazoku, the Maou is the absolute ruler of Makai and is acknowledged as the most powerful of them and typically makes their subjects know it. The Maou has numerous duties, most of which dealing with ensuring the safety of Makai as well as organizing attacks against Ningenkai. Usually a Maou claims their status either through birth, or overthrowing the current Maou using their unique talents. There is no higher honor, and many Mazoku would slit their own mother's throats to get it (and some cases DO if their mother is in the competition for it).
Famous Maou include the bluehaired enigma of Chrono Trigger (called Magus in the USA), Zetta from Makai Kingdom (Or as he describes himself, "Maaaaaaaouuuuuuuuuuuuuuu Zetta!" like a football announcer after a team scores a point), and Yuuri from the shonen ai anime Kyou Kara Maou. It should also be noted that not all Maou seem entirely sinister -- Maou Fodessy in Shuffle! seems outright warmhearted.
Anyways, Makai-based Aeternalae are accurately called "Demons" and do that whole invasion song and dance.