Short answers, pulled from the story.
A battle royal in boxing and wrestling is a multi-competitor fight where the last person standing or the one with the most wins is declared the winner. In professional wrestling, it involves four to sixty competitors inside the ring, with elimination by pin, knockout, or over-the-top-rope depending on the promotion.
New York's State Athletic Commission banned battles royal in 1911. The events had been popular from 1870 to 1910 as opening acts for boxing and wrestling shows, but were considered increasingly shameful and disreputable before the ban.
Jack Johnson, Joe Gans, and Beau Jack all began their boxing careers in battles royal. Winning these events gave aspiring fighters enough prestige to move into more respectable boxing matches.
The WWE Royal Rumble is an over-the-top-rope elimination match that begins with two competitors and adds a new entrant every two minutes, typically up to thirty total. A standard battle royal starts with all competitors in the ring simultaneously, whereas the Rumble staggers entries throughout the match.
Koushun Takami's 1999 Japanese novel Battle Royale and its 2000 film adaptation established the genre. The premise, in which a group of people is forced to kill each other in a large arena until one survivor remains, became the template for a wide range of manga, anime, television series, and video games.
The most prominent battle royale video games include PlayerUnknown's Battlegrounds (2017), Fortnite Battle Royale (2017), Apex Legends (2019), and Call of Duty: Warzone (2020). Garena Free Fire, Rules of Survival, Call of Duty: Black Ops 4, and Fall Guys: Ultimate Knockout are also widely played examples of the genre.