Increased Odds Methods

Masuda Method
The Masuda Method is a method of hunting Shiny Pokemon that was first introduced in Generation IV (DPPt, HGSS). This method involves breeding two Pokemon of different languages in order to increase the odds of the offspring being a Shiny Pokemon. The increase in the odds of hatching a shiny is varied by generation; according to Bulbapedia, the rates are as follows: Note that foreign language Pokemon obtained in-game from NPCs do not count towards Masuda Method.

Consecutive Fishing
Consecutive fishing, also known as chain fishing, is a method of Shiny hunting used only in Generation VI (XY, ORAS). It is not in Generation VII (SM, USUM) due to the different fishing mechanics.

This method involves using any of the three fishing rods available in the game to repeatedly fish in the exact same spot. As more Pokemon hooked by the rod are encountered and defeated/ran from, the chances of hooking a Shiny Pokemon increases.

The chain breaks under the following conditions: If a different Pokemon is encountered than the initial Pokemon pulled up, this does not break the chain. Additionally, rods can be switched between; and Pokemon can be healed, given items, and shifted around the party menu without consequence. Any other actions may be performed as well, so long as they do not cause any of the bulleted conditions above to be met which would cause the chain to break.
 * The player moves from the slot they were standing in.
 * The player does not successfully initiate an encounter when they cast their rod.
 * The player catches a Pokemon.
 * The player encounters a Shiny Pokemon.
 * The player battles a Trainer or uses any online features.
 * The player turns off the 3DS or entirely exits the game (such as soft-resetting).

Currently there is no confirmed increase rate for consecutive fishing, but a majority of reports of Shiny encounters through this method have reported between a chain of 50 and a chain of 200 encounters.

For a list of Pokemon that can be found through fishing in Generation 6, see the Consecutive Fishing List.

Friend Safari
The Friend Safari is exclusive to Pokémon X and Y. It is located in Kiloude City, which is unlocked after beating the Elite 4. It allows the player to face a wide variety of Pokémon, including starters and pseudo-legendaries, all with a higher chance to be shiny.

Unlike other methods, the Friend Safari doesn't require any general set-up except for having at least one Safari. The odds of finding a shiny in the Friend Safari is ~1/820, and ~1/586 with shiny charm.

For a full list of Pokémon encounterable in the Friend Safari, please refer to this list