Most important methods!
Masuda Method
The Masuda Method is a special method that is exclusive to hatching eggs. But not any normal Pokemon can be breeded for the shiny rates to increase. The Pokemon needs to be from a country with a different language of origin from yours. For more specific information, please check out Beacon of Nick's video here.
Poke Radar
The Poke Radar is an exclusive method only in Pokemon X and Y Games. While walking in a specific patch of grass around the map, you can use the Poke Radar to encounter the same Pokémon every single encounter. It is a form of chaining, and every time you encounter the same Pokemon, your chances of getting a shiny will increase. For more specific information, please check out aDrive's video here.
Horde Hunting
Horde Hunting is exclusive only to games susch as Pokemon X and Y. When you have a Pokemon in your party equipped with the move Sweet Scent, you will encounter hordes in the wild. These hordes typically consist of Pokemon of the same species, so your chance of finding a shiny will drastically increase. For more specific information, check out SmkGaming05's video here.
Chain Fishing
Chain Fishing is another exclusive shiny method that is exclusive to games like Pokemon X and Y. Much like the PokeRadar, you can use a Pokemon with the move suction cups to increase the chance of you finding the same Pokemon in the water. This will increase your shiny rates every time you encounter the same one. For more specific information, check out aDrive's video here.