From Attack of the Fanboy: “After youve inserted the Vault Relic into one of the obelisks on Sanctuary you activate Mayhem Mode. There are varying difficulties within Mayhem Mode, from Mayhem 1 all the way up to Mayhem 3. Each one of these modes will have boosts to enemy health, shields and armor, making it a lot tougher than the standard difficulty.”
Source: N4G PC How to Unlock Mayhem Mode – What is Mayhem Mode in Borderlands 3