The Nuzlocke Badge is a key item that is obtained from the player's mother in the Phoenix Iota after completing almost the entire game. Despite being called a badge, it has no real effect that would cause it to be considered another one of the 16 badges of the game.
The Nuzlocke Badge is essentially a trophy obtained for completing the entire game on Nuzlocke mode. As such, it is impossible to obtain if the player either did not activate it from the list of challenges at the start or lost the challenge earlier in the game. It is not known to have any practical use aside from bragging rights.
The owner of this badge made it to the Phoenix Iota on a Nuzlocke!