The original version of the twist was introduced in Survivor: Vietnam. Starting from Day 2, all remaining castaways are given the opportunity to mutiny from their current tribe, and join the other tribe instead. They do not have to give a reason publicly as to why they have decided to mutiny, although their tribe may be able to assume why they have made such a move. People who have decided to mutiny do have the option to mutiny back to their original tribe another day, however it cannot be the same day OR the day after. By this it means that if you decide to mutiny, you must stay in that tribe for two more days (changed in Last Leap so that you cannot swap back until a tribe swap occurs). A maximum of two people from each tribe may mutiny per-day, and you must be present at the challenge to mutiny.
In Survivor: Last Leap the rules to the twist were amended, so that castaways were unable to mutiny back until after a tribe swap mixed up the tribes. This change continued into Unfinished Business, with the addition of castaways picking which tribe they mutiny to because of the three-tribe format.
For full mutiny history, see the Tribe Mutiny page.