The rumors, first off, fly in the face of the ridiculous fan theories out there about the Joker in the new DC Extended Universe, which if anything lends these rumors some credibility. To set the stage, while Ben Affleck has admitted he's interested in directing a solo Batman movie, neither he nor Geoff Johns (or anyone else at Warner Bros) have admitted they're actually writing the movie together as of right now - so yes, this is the rumored plotline of an unconfirmed movie being written by unconfirmed/reported writers.
The general plotline, then, is rumored to follow the story of Red Hood and his return to Gotham City - and life. The Red Hood in question is Jason Todd, the second Robin who was beaten then blown up by The Joker in the story Death in the Family. In the comic books, he originally returned (look away, non comics fans, we don't want you looking down on us) thanks to a version of Superboy punching the walls of reality. His story has been refined, with him being saved from near death by the Lazarus pit, a concept now familiar to DC live-action fans thanks to Arrow.
The rumor of the film sees Todd return as Red Hood, but first pretending he's Batman and making his former mentor look like a criminal. The eventual end-game confrontation would include Batman, Red Hood, and Joker, much like in the "Under the Red Hood" storyline.
Nightwing, aka Dick Grayson, the first Robin, meanwhile, is rumored to also have a supporting role in the film - after a mention in Batman v Superman: Dawn of Justice.
The story of Jason Todd's return was loosely adapted as the plotline of Batman: Arkham Knight, the final chapter in the video game series by Rocksteady Studios, and has also been made as an animated movie called Batman: Under the Red Hood.
Todd was teased in the trailer for Batman v Superman, as a Robin costume is looked upon forlornly by Bruce Wayne (Ben Affleck), defaced presumably by Joker. Todd may also be getting a name drop, similar to Grayson, in the film.
The Untitled Batman Movie has no release date, but is rumored to be written by Affleck and DC Entertainment CCO (and comic book writer and writer on the CW DC TV shows) Geoff Johns, with Affleck also targeted to direct.