Credit to PianoGeek for finding the source, lol.
The Nuzlocke Challenge is a set of rules intended to create a higher level of difficulty while playing the Pokémon games. Many challengers feel that the rules also serve the purpose of encouraging the use of Pokémon the player would not normally choose, and promoting closer bonds with the player's Pokémon. The rules are not an in-game function, but are self-imposed on the part of the player, and thus subject to variation.
The name of the challenge originates from the comic series of the same name, which features a Nuzleaf resembling Lost character John Locke as a recurring gag character.
The Nuzlocke Challenge has only two rules that must be followed:
Any Pokémon that faints is considered dead, and must be released or put in the Pokémon Storage System permanently (or may be migrated or transferred with Poké Transfer, as long as the Pokémon is never able to be used again during this run).
The player may only catch the first wild Pokémon encountered in each area, and none else. If the first wild Pokémon encountered faints or flees, there are no second chances. If the first encounter in the area is a double battle, the player is free to choose which of the two wild Pokémon they would like to catch but may only catch one of them. This restriction does not apply to Pokémon able to be captured during static encounters, nor to Shiny Pokémon.
Other near-universally used rules include:
The player must nickname all of their Pokémon, for the sake of forming stronger emotional bonds.
The player may only use Pokémon they have captured themselves, meaning Pokémon acquired through trading, Mystery Gifts, etc., are prohibited. As for trading and retrading the same Pokémon (for the purpose of evolving a Graveler, for example), there is no firm consensus. As of White: Hard-Mode Episode 3, it is implied that the player can accept Pokémon that are received freely from NPCs.
The player may not voluntarily reset and reload the game whenever things go wrong. Being able to do so would render all of the other rules pointless.
Ok, those are the basic rules that basically everyone follows. Now the ones that can be used to increase / decrease difficulty.
The player's Starter Pokémon must be randomly chosen. A common system is if the last digit of the player's Trainer ID number is 1-3, the player must choose the Grass-type starter; if it is 4-6, the Fire-type starter; if it is 7-9, the Water type starter; if it is 0, free choice. Alternatively, use the Trainer ID modulo 3 for the same purposes.
A black out/white out is considered to be a "game over," even if there are usable Pokémon left in the PC, and the player must start over.
The player may only catch the first Pokémon after each Gym battle instead of in each area.
The player must use the same number of Pokémon as the opponent uses during a Gym battle or rival battle.
The battle style must be changed to "set" in the options menu, meaning the player does not get the opportunity to switch out their Pokémon after an opponent's Pokémon faints.
The player's Starter Pokémon must be released or permanently put into a PC box after the first wild Pokémon is caught.
Potions and status-healing items may not be used, so the player may only use Pokémon Centers for healing.
Or, Pokémon Centers may not be used, meaning only Potions and items may be used for healing.
The player is limited in their Pokémon Center visits to a certain number per town.
Held items may not be used.
The number of Poké Balls able to be purchased per Poké Mart is limited to a certain number.
Poké Marts may not be used; the only items that may be used are those found in the overworld or given to the player by NPCs.
Master Balls may not be used.
The player may not evolve their captured Pokémon, but evolved Pokémon may still be caught.
(Black 2 and White 2 only) The difficulty must be set to Challenge Mode, which increases the levels of opposing Trainers' Pokémon.
Legendary Pokémon may not be used.
The player may not flee from battle.
The player may not use Pokémon above a certain level limit based on the level of the next Gym Leader/Elite Four/Champion's highest leveled Pokémon. What to do with Pokémon in a player's collection that surpass the level limit is up to the player.
Poké Balls may not be used. Any Pokémon obtained must be either given to the player or hatched from an Egg.
The Day Care may not be used.
The Exp. Share may not be used.
Quality-of-life features, such as Pokémon-Amie, the DexNav, or Super Training, may not be used.
Online resources (walkthroughs, guides, etc.) may not be used.
The core rules are not in effect until the player has gained their first Poké Balls and thus the ability to catch Pokémon. For example, encounters starting from the PoochyenaRSORAS/ZigzagoonE that the player has to save Professor Birch from, up to when the player has the ability to catch Pokémon, are not counted. Likewise, in the games where the rival battle is immediately after obtaining the starter Pokémon, the "any Pokémon that faint must be released" rule is often not enforced at that time.
Species/Dupes Clause: The player may continue fighting Pokémon in a new area until a species is encountered that has not been caught yet, which then immediately counts as the first encounter. This is to increase variety in a player's Pokémon collection.
A limit may be set on how many times they can apply the Species/Dupes Clause in an area. If this many duplicate Pokémon are encountered in an area, the Species/Dupes Clause is no longer applied for that area and the player has to settle for the next Pokémon they encounter, regardless of its species.
The player may have a small number of "second chances" or revives of fallen team members.
Shiny Clause: Shiny Pokémon do not need to be released if they faint.
Each Gym Badge may act as a checkpoint. If the player gets a game over, they may start over from when they got their previous Gym Badge.
If the player has no Pokémon that can use a field move that is required to continue the game, they may catch another Pokémon that can learn said field move. However, it cannot be used in battle for any reason, and must be released, permanently put into a PC box, or migrated as soon as it is no longer needed or if the player catches another Pokémon that can use said field move.
The "first encounter only" rule is modified for within the Safari Zone. One encounter may be had for each area, or one encounter may be allowed for the entire Zone.
Then there's a variation of a Nuzlocke listed called a Wonderlocke.
This is a more unpredictable variant of the Nuzlocke Challenge. In this variant, any Pokémon caught must immediately be traded using Wonder Trade and the received Pokémon should be used instead. Typically this comes with a level restriction where if the received Pokémon is more than a certain number of levels higher than the original Pokémon, it must be traded again until an appropriate-level Pokémon is received.
Hope this helps! :)