I've never heard of it. What is it?
Apparently, how D&D started. The original club were all Napoleonic wargamers who came up with the idea of campaign play - instead of just playing individual battles, when a battle went differently from its historical version, they would play out how the war went. They used a book called Strategos, which was a US Army training exercise from the 1800s, which included the fateful line that the Referee should "bear in mind the principle that anything may be attempted," and because their game had things like the fog of war, they would do things like garrison in towns and ask the civilians for intel.
One of them, David Wesley, had an idea for an experiment and told the group that he wanted to try something for their next campaign, that they were to play a battle over this bordertown named Braunstein, but had to play out a short little improv session to see what the disposition of the town was. Wesley had no intention of actually playing the battle, he was interested in running this weird social game, where one player was the Prussian Landwehr Commander, one player was the Mayor, one player was the Baron, one player was the Chancellor of the University, etc.. A lot more people showed up than Wesley anticipated, so he just started throwing out roles like Tavern Owner, Other Tavern Owner, various students at the University, a French spy, etc.. Wesley laid out the table for the town of Braunstein, assigned everyone a figure and a narrative objective based on their role, and cut them loose. Anything they wanted to "do" went through him, but otherwise they wandered around the town talking to each other. Famously, Dave Arneson got into an argument with one of the other players, challenged him to a duel, and ended up the first person ever killed in a roleplaying game. This idea of each player controlling one figure instead of an entire army and solving problems by exploring, fighting, and talking is generally considered the birth of the tabletop RPG.
Wesley thought that Braunstein was an utter failure because the players mostly just talked to each other and didn't run stuff by him, so at the end he didn't have a full picture of what had happened. His players loved it, however, and he ended up refining the formula. "Braunstein IV" was set on a fictional banana republic ('Banania') with different players taking the roles of powerplayers as the government collapses - four different military commanders, government officials like the Treasurer and Head of the Secret Police, local business owners, local union reps, revolutionary groups, etc.. This time, a lot of the roles had control of a small number of figures (the army commander controls a couple of soldiers, for instance, or the union guys have groups of workers, the revolutionaries have guerrillas, etc.). There are a bunch of videos out there about it, none of which are perfect, but
Questing Beast's is decent.
Modern Braunstein play isn't really formalized, but I've heard it simplified as much as "multiple actors operating in conflict under a fog of war." To me, it's just letting your players actually be independent actors, transforming the GM from storyteller back into referee. Everyone tries to make their campaign Lord of the Rings or Star Wars (and usually fails), but I think if you slip the reigns a bit, let your players control their own factions (or be a faction of one, however difficult), it opens you up to stories like Game of Thrones or Succession - and ultimately requires a lot less work on the GMs part, as your players are the ones actively making the story.
The BrOSR's "Brozer" Braunstein is...kind of really dumb, but there's some remarkably interesting ideas in it, imo.