The term Masuda method is named after Game Freak director Junichi Masuda. Masuda first programmed the method into Pokemon Diamond/Pearl to help Pokemon fans have an easier time of collecting Shiny variations of their favorites.
That’s not to say you’ll be swimming in Shiny Pokemon if you adhere to the Masuda method. Your chances of obtaining a Shiny Pokemon in the wild are about one in 8192, whereas your chances of breeding one via the Masuda method is five in 8192 (Generation IV), or three in 4096 (Generation V).
For the Masuda method to work, you need to breed two Pokemon of different real-world geological origin. For example, a Pokemon from the United States, and a Pokemon from Japan. Trading Pokemon with trainers worldwide is much easier than it used to be, thanks to the Wi-Fi capabilities of the Nintendo DS and the Nintendo 3DS.
There are some restrictions to the Masuda method. Foreign-language Pokemon that are obtained through in-game trades are still considered to be from the same country as the game they were generated in, so those Pokemon won’t stand a higher chance of yielding Shiny offspring. A good example is Lt. Surge’s French Pikachu, Volty, in Pokemon HeartGold/SoulSilver. Volty technically doesn’t count as a foreign Pokemon, and the Masuda method therefore won’t take effect.
Shiny Pokemon are not necessarily better or stronger than their regular counterparts. Their altered appearance simply makes them a novelty, and catching or hatching one is a point of pride because they’re so rare. In a way, Shiny Pokemon can be regarded as collector’s items.
Masuda outlines the Masuda method on his blog, and Bulbapedia goes into detail about the calculations.