This page explains what shiny pokémon are and some methods for improving your chances of getting them.
What are Shiny Pokémon?
A shiny pokémon is a pokémon with different coloration from the usual. Being shiny only affects the pokémon's appearance. It does not affect its stats, moves, breeding results, or anything other than its appearance.
Finding Shiny Pokémon in the Wild
In most cases, finding a shiny pokémon in the wild depends on luck, and is extremely rare. Before Pokémon X and Y, there was a 1 in 8192 chance of encountering a shiny pokémon. Since X and Y, there is a 1 in 4096 chance of encountering a shiny.
Starting in Pokémon Black 2 and White 2, it became possible to obtain the Shiny Charm, an item that makes you three times more likely to encounter shiny pokémon, giving you a 3 in 8192 chance in games before X and Y, and a 3 in 4096 chance in X and Y, and subsequent games.
To obtain the Shiny Charm, you must catch all of the pokémon in the pokédex in your game, excluding certain event pokémon. The Shiny Charm is not available in games released before Black 2 and White 2.
Junichi Masuda, who has worked on all of the Pokémon games thus far in varying roles, such as composer, designer, programmer, etc., added a method to Pokémon Diamond and Pearl to increase the chance of hatching a shiny pokémon from an egg. This method is known as the Masuda Method, and has been included in all Pokémon games since Diamond and Pearl.
To use the Masuda Method, breed two pokémon that each come from different real-world regions (e.g. an American pokémon and a Japanese pokémon). The resulting egg is more likely to produce a shiny pokémon.
Using a shiny pokémon to create an egg will NOT increase your chance of getting a shiny egg.
How Do I Know if my Pokémon is Shiny?
Shiny Pokémon will be a different color from the usual color for that type of pokémon. However, sometimes the color difference is subtle.
When a shiny pokémon appears in battle, it will make a sparkling sound after it makes its usual pokémon sound, and stars will appear around the pokémon for a moment.
If you have a pokémon in your party or box and you are not sure if it is shiny or not, look at the summary screen for the pokémon. If there is a red star somewhere on the summary screen for the pokémon, it is shiny.