First, you need to start a new game. The basic rule of a Nuzelocke are you can only catch the first Pokemon you encounter on a route. If you fail to catch it, you get no new Pokemon for that route. If a Pokemon faints, it is considered "dead" and can no longer be used. You can't use your starter once you can start catching other Pokemon. Wonderlocke takes it a step further. When you catch your Pokemon for a route, you then Wonder Trade it and have to use that Pokemon. If you get a duplicate, you can Wonder Trade it again. Some people add other rules like being able to Wonder Trade again if you get a Pokemon that is too high a level to obey, or not being able to use experience share
Source: https://gamefaqs.gamespot.com/boards/210930-pokemon-ultra-sun/76727626
For the variations, there are
Wedlocke: Pokémon are assigned into pairs or “couples” by gender. Couples fight together, and when one member of a couple enters a battle, the player may only switch between that Pokémon and its partner for the duration of the battle, or until both partners are knocked out
Monolocke: Only Pokémon that have a certain type may be caught and used in the player’s team. Sometimes referred to as the “gym leader challenge”
Apocalocke: The player selects (or is assigned) an apocalyptic disaster as the theme for the run. Depending on the disaster chosen, only certain types of Pokémon may be caught and used
Giftlocke: The player is not allowed to catch Pokémon, and must use only their starter and gift Pokémon that are obtained without catching
Uniquelocke: None of the player’s party Pokémon may share a type
NOHKOlocke: A player’s Pokémon must be boxed or released if it knocks out an opposing Pokémon from full health in a single hit
Chesslocke: The player may only capture a certain number of Pokémon, each of which must be assigned a role and certain restrictions based on various chess pieces
Zombielocke: Pokémon that are knocked out may be revived as “zombies”, and may only be healed by sacrificing other Pokémon
Loserlocke: The player may only use Pokémon from a one or two-stage evolutionary line, and may only use regular Poké Balls
Starlocke: The player begins the game with one Pokémon of each type (eighteen total), and may not acquire any more. They may not switch out party members unless fainted
Sleeplocke: The player must complete the run in a single sitting, without falling asleep in real-life
Tasklocke: Intended to boost the player’s real-life productivity. The player must nickname their Pokémon after real-life tasks and chores which must be completed before major battles
Draftlocke: Multiple players race to complete a Nuzlocke challenge. At the start of the game, players take turns drafting types, and may only use Pokémon of the types they’ve drafted
Source: https://nuzlockeuniversity.ca/nuzlocke-variants/
Finally, here is a link explaining what an egglocke is
https://pokemondb.net/pokebase/286561/what-is-an-egglocke
