Divine Gate Collaboration Event - Puella Magi Wiki
Here's more information. So basically Pia is a wind demon from another setting, and she wished to make friends, which apparently was powerful enough to cause a crossover false world that pulled in the five main girls. It's basically Alt!Rebellion, as Pia feels kinda bad about it and tries to send everyone home.
Each of the girls is associated with an element: Madoka with void, Homura with darkness, Sayaka with water, Mami with light, Kyoko with fire, and Pia with wind.
Madokami happens when Madoka wishes to save everyone, uber-attacking Divine Gate and resetting the multiverse, but not after comforting Pia and everyone assures her they're friends forever.
Madoka the Roaring Beast is associated with a story snippet in which Madoka resonates with the demon world and transfigures into one instead of becoming a Magical Girl, apparently the failure state if Pia doesn't destroy the crossover false world. Her ascension to demonhood is celebrated by an unnatural rainfall resonating with a kind and gentle feeling. The Netherworld becoming just a little bit less brutal...
There's also a Homura "Dark Love Fairy", image spoilered below.
Another failure state character. Seeing Madoka being acclimating to the Demon World causes Homura to basically go "You are worth Hell" and throws herself fully into it, apparently a parallel path to Homucifer. She gains control of the Crossover World and doesn't let anyone destroy it, cherishing her new heaven and hell with Madoka.