'Win the Button' is a tournament where the player who wins the last pot receives the dealer button for the next hand.
This means you can get the button way more often than in a regular game. But it means that if the players next to you keep winning pots, you might find yourself in the blinds more often too.
In some tournaments, the format switches back to normal button movement later on.
For full details for a specific event, check out the tournament lobby.