From Bulbapedia, the community-driven Pokémon encyclopedia.
An alternate colored, or shiny, Pokémon is one that has a color different from the natural one featured in the Pokédex. For example, a shiny Zubat would be green instead of blue. Shiny Pokémon are rarer than their normally-colored counterparts, and follow no set pattern in their color change per species. However, all shinies of a species have the same coloration. The probability of getting a shiny in any game is 1/8192.
Shiny Pokémon are only in games from Generation II onwards, though Pokémon originally obtained in a Generation I game can be shiny if moved elsewhere, if their stats match up.
In the games, when a shiny Pokémon enters a battle, stars flash and surround it. Also, three stars are indicated in a shiny Pokémon's Summary screen.
In Generation II games, alternate coloration is determined by its IVs, with Speed, Defense, and Special being 10, and Attack being either 2,3,6,7,10,11,14 or 15. In 3rd generation games, shiny Pokémon aren't determined by genes, but by its Personality value
Also in Generation II, there is a shining Gyarados at the Lake of Rage, which will give the player the Template:I when captured or defeated by the player.
In the TCG, shiny Pokémon are called "shining" or "star" Pokémon.
In the anime, only three concrete examples of shining Pokémon have appeared: Ash's Noctowl, the aforementioned red Gyarados, which Lance caught, and Winona's Swellow. Similarly to the games, they sparkle a bit when released from their respective Poké Balls.
Many glitch Pokémon encountered in Generation III or Generation IV will be shiny upon entering battle. This may be due to their already strange stats.