The Nuzlocke Challenge Rules:

Encounters: The player may only catch the first wild Pokémon encountered in each area, and no others. If the first wild Pokémon encountered faints or flees, there are no second chances.
If the first encounter in an area is a Double Battle or Horde Encounter, the player may choose which of the encountered Pokémon to catch, but only one of them.

Dying: Any Pokémon that faints is considered dead and must be released. If you run out of living Pokémon, you've failed the challenge and must restart the game.

Shiny Claus: Any SHINY pokemon encounter, regardless of if you already captured a pokemon in that area, may be caught.

Nicknames: Nicknames are mandatory (unless the seemingly crash the game, haha, forshadowing).

Species/Dupes Clause: The "first wild Pokémon in each area" rule does not apply in an area until a species or evolution line is encountered that has not been caught yet. For example, if the player's first encounter in an area is with a Caterpie and they already own a Caterpie, Metapod, or Butterfree, it wouldn't count as their first encounter in that area. This allows for increased variety in a player's Pokémon collection.