Shiny Pokemon
First introduced in Pokemon Gold, Silver and Crystal, a Shiny Pokemon is a rare form of Pokemon that differs in color from a normal Pokemon. It is easily identified by the red star to the right of its sprite on the summary screen. The difference can be subtle (Shiny Leafeon are only a slightly lighter shade of teal) or dramatic (Shiny Gyarados are a brilliant shade of red, as opposed to normal blue). When they emerge from a PokeBall in battle they're accompanied by a brief glittery animation that makes them appear to shine.
Besides being a different color than a normal Pokemon, Shiny Pokemon are exactly the same. Contrary to popular belief, Shiny Pokemon don't have higher stats (HP, Attack, Defense, etc.) than normal Pokemon. So why would you bother to catch a Shiny Pokemon? Basically, it's like a trophy to show off to your friends.
As mentioned, Shiny Pokemon are very rare, but how rare are they? You have a 1 in 8,192 chance of encountering a Shiny Pokemon in the wild - that's a negigible 0.0122% chance of running into one! The chances are just as dismal if you try to breed for a shiny egg.
So what happens if your Shiny Pokemon evolves? Don't worry, your luck won't run out as Pokemon remain shiny, even after they evolve. The same goes for Pokemon transferred through Pal Park - if they were shiny in the 3rd Generation, they will remain shiny in Pokemon Diamond and Pearl.
Pokemon Gold, Silver, and Crystal
In the 2nd Generation, Pokemon shininess was determined by Individual Values (IVs) - hidden stats that determine the base value of the stats of your Pokemon. Pokemon with Speed, Defense, and Special IVs of 10, and an Attack IV of 2, 3, 6, 7, 10, 11, 14 or 15, will be shiny. Due to their higher IVs than normal, Shiny Pokemon have only slightly above average stats. To find out if your Pokemon is shiny, look at its stats and if it has 3 pluses by its gender, then it's a Shiny Pokemon.
The most famous example of a Shiny Pokemon is the Red Gyarados that lives in the Lake of Rage. The Red Gyarados is the only Pokemon in the games that will be shiny 100% of the time when encountered.
Pokemon Crystal
When you first visit the Daycare Center below Goldenrod City, the Daycare couple will give you the Odd Egg. It takes up a Pokemon slot just like a normal egg. It can hatch into one of seven Pokemon: Magby, Smoochum, Pichu, Igglybuff, Cleffa, Tyrogue, Marill or Elekid. What makes the Odd Egg special is that it has a 50% chance of hatching into a Shiny Pokemone. The hatched Pokemon will always be female, with the exception of Tyrogue, and will also know Dizzy Punch.
Pokemon Ruby, Sapphire, Emerald, FireRed, and LeafGreen
In the 3rd Generation, Pokemon shininess was no longer determined by IVs. Instead, the two trainer IDs (the one on the Trainer Card and the unknown trainer ID) make up two of the four random numbers. While these numbers remain constant throughout the game, the remaining two random numbers are determined each time you run into a wild Pokemon, hatch a Pokemon egg, or are given a Pokemon (such as a starter Pokemon). Since shininess is no longer determined by IVs, it is now possible to have a Shiny Pokemon with maximum stats.
Pokemon Diamond, and Pearl
In the 4th Generation, for the first time ever there is now a way to increase the chances of encountering a Shiny Pokemon. Using the PokeRadar you can create a Chain. The longer your current PokeRadar Chain is, the higher your chances of encountering a Shiny Pokemon. Shiny Pokemon are only found in patches of grass that glow white twice when using the PokeRadar.