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So BL2 showed a bunch of vaults on other planets. BL3 is the Vault Hunters/Crimson Raiders trying to get their hands on those vaults before anybody evil does.
BL3 shuffles the players between three planets, each one with it's own Vault and it's own conflict. Each one would presumably have a different environment or somthing as well.
The first planet is Pandora. When Handsome Jack died, a whole mess of Hyperion troops and equipment were left behind. Jack's crusade had already caused Bandits to start gathering in larger numbers, now the bandits are resurgent.
The Crimson Raiders on Pandora are soldiers: Former Crimson Lance and Hyperion for the most part, led by Axton, Mordecai, and Zer0.They fight alongside the Players against the Bandit hordes.
The second planet is largely urban. A Vault has opened up there, and the Tediore corporation is trying to take control of it. Legions of smiling troops and robots are marching against the Crimson Raiders. On this planet, the Raiders are Bandits, The Slabs plus others, led by Salvador, Brick, and Kreig.
The third planet is a jungle. A hyperion mad scientist escaped with a bunch of iridium and animals from Pandora. He cloned some Pandora beasties and mutated local animals and is using them as a personal army. Here, the Crimson Raider forces consist of an endless wave of Robots designed by the combined minds of Gaige, Tiny Tina, and MISTER TORGUE FLEXINGTON (Former CEO of the Torgue corporation, they fired him).
Maya and Lilith are off searching for information about the Vaults/Serving as reinforcements wherever needed.