At first let me "warn" you: Only attempt this Nuzlocke if you know what comes
You are only allowed to catch the first Pokémon you meet in each new area, if it faints or flees, no second chances
Any Pokémon that faints is considered dead, which means it has to be either released or put in the Pokémon Storage System permanently
The general consensus is that players must also nickname all of their Pokémon, for the sake of forming stronger emotional bonds.
The general consensus is that a black out/white out is considered to be "game over", even if there are Pokémon left in the PC, exceptions can be made for one specific battle only
Strongly implied is the stipulation that the player can use only Pokémon they have captured themselves, meaning traded Pokémon, mystery gifts, etc., are prohibited. Trading and retrading the same Pokémon (for the purpose of evolving a Graveler, for example) is something of a gray area, and may fall under optional rules. it is implied that the player can accept Pokémon that are received freely from NPCs.
Also strongly implied is a prohibition against voluntarily resetting and reloading the game when things go wrong. Being able to do so would render all of the other rules pointless.