The Masuda method , seemingly named for Junichi Masuda, a person involved in the development of the Pokémon series, has one notable benefit: Shiiiiiiiiiiiiiiiiny Pokémon! Shiny Pokémon are not extremely notable on a level of depth. Their movesets and the like are the same as their non-Shiny counterparts.
But the one thing that makes them so desirable is their coloration. Pokémon often look different when Shiny - a Shiny Gyarados is red instead of blue, a Shiny Sceptile is cyan instead of green, a Shiny Moltres is pink instead of yellow/red, a Shiny Salamence is green not blue, a Shiny Rayquaza is black instead of green, etc.
Under most circumstances, Shiny Pokémon are extremely rare. The odds of finding them are 12 in 65,536, commonly rounded to 1 in 4,096, about 0.0244% .
Think about that for a moment. You might go through 500 wild battles in a relatively completionist playthrough X/Y, yet that’s barely 1/10 of the way to the 1 in 10,000 a Shiny Pokémon will appear in. Normally, finding Shinies requires tedious looking in an area.
Additional methods to increase the rate of Shinies have been brought in over the years - like forced encounters (Shiny Gyarados in Gold/Silver/Crystal), special items (Shiny Charm, Black 2/White 2, tripled rate), and PokéRadar chaining (DS Pokémon games, 40x more effective at best).
Of course, all of them still require some level of work. The one with the least true “work” involved is the Masuda method of Pokémon breeding. In it, you will need two Pokémon, each from a different region - that means you can have one of your own, and one from elsewhere.
The region is denoted on the Pokémon’s status screen by some form of abbreviation where yours are normally blank. (Plus, Japanese/Korean characters are a dead give-away. =P)
Basically, you breed the two together … and that’s it. There’s nothing special involved with this method that you don’t want to be - if you’re just looking for a Shiny, without desiring certain IVs or moves or nature or whatever, you can just breed them, but this otherwise still falls under normal breeding doctrines.
The chance of finding Shiny Pokémon like this is 1 in 1,366 - about three times more effective than random hunting/breeding. It’s not much, but it’s practically the only way to, for example, get Shiny starter Pokémon!
Keep in mind Wonder Trade for this - lots of people use that. U.S. players can expect to get foreign Pokémon 40% ~ 50% of the time, which means residents of other regions are very likely to get something foreign (if just because the U.S. dominates half the market).