Shiny Pokémon

Shiny Pokémon are very rare forms of Pokémon that are different in color when compared to other Pokémon. The term 'shiny' is a reference to the sparking animation and sound effect when the battle starts. The term "shiny" in itself, however, was unofficial prior to Generation V, when the Pokédex began cataloguing shiny Pokémon and using the term itself.

The game has a way of deciding if a Pokémon is Shiny. In Generation II, when the Battle starts, the game picks four random numbers between 0 and 65535. If any of the numbers is less than eight, it is a shiny Pokémon. The chance of seeing one of these is about 1 in every 8192, or 0.0122%.

From Generation III onwards, shiny Pokémon are determined by other factors such as the Trainer ID number and the personality value of the Pokémon.

Methods
There are a few different methods to increase your chances of obtaining a shiny Pokémon.

Chaining (Gen IV)
This method uses the Pokéradar to encounter chains of the same Pokémon. Here's a few tips:
 * If the bush just shakes, it is a Sinnoh Pokémon.
 * If the bush has a whitish shake, it might be a non-native Pokémon.
 * Never use it in water, caves or tall grass.
 * The bush with the same type of shake as the first Pokémon you battled that is the farthest away within a four by four grid most likely is the same Pokémon.

Breeding (Gen IV-V)
If you breed two Pokémon from two different language cartridges, the chance of an egg hatching into a shiny Pokémon increases. For example, a Pokémon originating from a Japanese game and a Pokémon originating from an American game will have an increased chance of producing a shiny offspring.

Soft Reset (Useful for Legendaries and Shiny Starters (Gen. II to Gen. V))
One method which is easy enough (but often very time consuming) works for starters and most Legendary Pokémon. Basically, the method is to stand in front of the legendary Pokémon you will catch/starter you will take and save. If you don't get a shiny when you get your starter/battle the legendary, soft reset the game by pressing A+B+Start+Select on the GBA and L+R+Start+Select on the DS. This method can easily require over 1000 resets before you get a shiny, however with enough dedication and patience your efforts can pay off.

In-Game Shiny Pokémon
In-game shiny Pokémon are met as part of the main story in some Pokémon games. For example, the Red Gyarados appears in Pokémon Gold, Silver, and Crystal Versions, and in Pokémon HeartGold and SoulSilver Versions as part of a story arc, and can be caught in a way similar to most legendary Pokémon.

In FireRed and LeafGreen, a Trainer with a shiny Espeon can be battled inside the Trainer Tower on one of the Sevii Islands.

Shiny Pokémon in the Anime series

 * Noctowl: Ash Ketchum caught a Shiny Noctowl while on his travels through Johto. It also seems to be about half the size of other Noctowls.
 * Shuckle: While in Johto, Ash and co. discovered a Shiny Shuckle.
 * Gyarados: First seen in the Lake of Rage and was caught by Lance. It was also seen again during the battle between Kyogre and Groudon.
 * Magneton: Owned by Pokémon Trainer Jackson when he battles Ash in the Silver Conference.
 * Swellow: Winona, the Fortree City Gym Leader has a Shiny Swellow.
 * Donphan: A Shiny Donphan has been shown in the Hoenn Region on Dontoe Island.
 * Magikarp: Was seen briefly in the episode "Judgement Day".
 * Dustox: Jessie's Dustox fell in love with a male Shiny Dustox and then was released similar to how Ash released his Butterfree. (Except Jessie smashed Dustox's Poké Ball so it could go with it's true love)
 * Metagross : In the episode "Noodles! Roamin' Off", James in Team Rocket went out to see this strangely colored Metagross that people had been talking about, and was promptly attacked by the shiny Metagross.
 * A Ditto in "Dealing With Fierce Double Ditto Drama" was shiny.
 * Ariados: In the Special Episode "Dawn's New Journey", Dawn and Cinco encountered a swarm of Ariados as well as a shiny one using Psychic so the others could wrap Dawn and the others with String Shot.
 * Entei, Raikou, and Suicune: the three legendary beasts were shiny guardians of the city in the Pokémon Movie: Zoroark: Master of Illusions

Trivia

 * In Generation II games shiny Pokémon are, on average, slightly stronger than their normal colored counterparts. However, this is not true for later generations.
 * If you evolve a shiny Pokémon it still remains shiny.
 * There are examples of Pokémon which, although coloured differently, are not officially shiny (as their colouration differs from the shiny variant).
 * In Pokémon Mystery Dungeon: Explorers of Time & Explorers of Darkness, there is an alternately colored Celebi.
 * The first alternately colored Pokémon to appear in the anime series was seen by Ash in the first season when he released his Butterfree to mate with it.
 * In Pokémon Mystery Dungeon: Red Rescue Team and Blue Rescue Team, two Kecleon own a shop in the town square. One is a dark purple color.
 * The Pokémon which reside in parts of the Orange Islands are coloured differently to other Pokémon. The colour changes are due to climate and natural adaption according to Professor Ivy.