JOKER
The "White Joker"
When Jun Kurosu is taken over by Nyarlathotep's power, he becomes known as the Joker. The Joker uniform is a kind of persona that Jun can summon, possibly a form of the persona called JOKER-NYARLATHOTEP, which has the two battle forms pictured on this page (angel and devil). Supposedly you could influence the form you would battle by spreading rumors that the Joker was like an angel or like a devil.The Joker costume is a black-and-white uniform, very similar to a school uniform, and Jun wears an odd, stylized mask on his face. Part of it obviously hearkens back to the Joker in Batman - the insane smile, for sure. The black shadow over half of its face matches the hair that falls over Jun's eyes, and the weird jester's cap matches the Joker on playing cards. The weird Pharaonic beard on the mask - I'm tempted to guess that it comes from H.P. Lovecraft's story "Nyarlathotep," where Nyarly appeared in Egypt, looking like a pitch-blach Pharaoh. (He had no hair or beard, but the idea is there...and Pharaonic beards were fake anyways.)
The Joker is the head of the Masquerade, an organization devoted to bringing humanity to the utopia called Idealian. To do this, the Joker spread a rumor that if you called your own cellphone number, the Joker would appear and make your dreams come true. In reality, the Joker would appear with a crystal skull and absorb the caller's "Ideal Energy" - the energy that they devoted to their dreams for the future - and stored the energy to raise Sumaru City towards Idealian. Nyarlathotep also warps the Joker's memories of his childhood to give him incentive to keep going.
The "Curse Joker"
It's important to make the distinction between the White Joker (Jun), and the Curse-Joker (Tatsuya Sudou) and the Black Jokers of Eternal Punishment.The White Joker was Jun, who fundamentally was a kind person who wanted people's dreams to come true. So to summon him, you performed the "Joker CHARM" to make your dreams come true. The word "Joker" was also written in katakana: ジョーカ様
The Joker you meet at the beginning of Eternal Punishment clearly could not be Jun - at least to anybody who's played Innocent Sin. Jun would never spread a rumor that if you summoned him, he'd kill the person you hated the most. That's more along the lines of Tatsuya Sudou, King Leo. If you'll notice, the paper bag that Leo wears over his head is a crude replica of Jun's Joker mask; the insane smile and a single visible eye! You perform the "Joker CURSE" to get this murdering Joker, and his name is written in Roman letters, all-caps: JOKER.
Black Jokers
People turn into Jokers in Eternal Punishment. Again, their faces closely resemble Jun's mask - pale, with a wide smile and only one eye visible. This is called the "JOKER-form" or "JOKER-transformation" (JOKER-化).These Black Jokers, the personae of people who have become JOKERS, are forms of Nyarlathotep, manifestations of their "sin." As Nyarlathotep's power increases, the bigger and bigger the Black Jokers get.
"Joka" 女媧
There's also the "Joka" mentioned by Wang Long Chizuru, who calls it a dragon of misfortune. This is written in kanji because it's actually the name of the Chinese mother-goddess Nuwa, who created humanity out of mud. Many other stories mention this Joka - Houshin Engi, for example - and you can see the statue of Joka at the gates of Utena in Sumaru Castle. She was half-human, half-dragon/snake.Other fun notes about this stuff: the "Kotodama" (言霊) mentioned repeatedly in Eternal Punishment literally means "The Spirit (Power) of Words." It's basically the same as "Rumors." So when Tatsuzou performs the "Ritual of Kotodama," he's more or less bringing Gozen back to life with the power of rumors. "Kegare" (穢れ) is essentally "impurity" or "sin" = "Tsumi."