A box legendary is the Legendary Pokémon featured on a main game's box art, which the player encounters as part of the story and can usually catch before the credits, with each paired version getting its own. Ho-Oh and Lugia, Groudon and Kyogre, and Koraidon and Miraidon are box legendaries.
The term is fan shorthand built from how the games are sold. Since Pokémon Gold and Silver put Ho-Oh and Lugia on their covers, the paired versions of each generation have each carried a flagship Legendary, and meeting it is the story's climax rather than an optional hunt. In Generation III you stop Groudon or Kyogre's rampage, in Generation IV you face Dialga or Palkia at the Spear Pillar, and in Generation V the plot forces a battle with Reshiram or Zekrom before the final fights and expects you to catch it on the spot.
Box legendaries usually have a base stat total of 670 or 680, sit at the top of their regional Pokédex story, and are version exclusive by definition: your copy of Violet will never let you catch Koraidon, because Koraidon is Scarlet's. Trading between versions is the designed answer. Generation IX made its box legendaries partners from the opening hour, since Koraidon and Miraidon serve as your ride Pokémon for the whole game before their battle form unlocks.
Not every Legendary is a box legendary. The roaming beasts, the Regi group, and event Mythical Pokémon sit outside the cover system, and some games add a third flagship, such as Rayquaza for Emerald or Giratina for Platinum, on the enhanced version's box.
In play"The box legendary in this version is the dragon, so that is the one you will ride around on."