Shiny Pokémon

Whenever you encounter a Pokémon, it has a very small chance of being a shiny. Shiny Pokémon have different colors than normal Pokémon and also sparkle when they enter battle. However, they are extremely rare. You only have a 1/8192 chance of seeing a shiny Pokémon. Shinies can appear anywhere and everywhere. They can even appear in places where you can't catch them, such as in the Battle Tower:


A Shiny Manectric! Oh darn, it's in the Battle Tower, so it can't be caught. =(

Some notes on Shiny Pokémon:
*A shiny Pokémon will still be shiny after you evolve it.
*Breeding a shiny Pokémon won't guarantee a shiny baby.
*If you trade a Pokémon from Red/Blue/Yellow to Gold/Silver/Crystal, there's a chance it will be shiny even though shiny Pokémon don't exist in RBY.
*The Red Gyarados from Gold/Silver/Crystal is the only guaranteed shiny you'll encounter in the games.

Easy Shiny Pokémon to Catch

Well, the term "easy" is misleading because you still only have a 1/8192 chance of getting a shiny Pokémon. But the easiest shiny Pokémon to get are the event Pokémon. These include Pokémon like the starters, Eevee, Beldum, Castform, and legendary Pokémon that don't move around.
For the starters and other other event Pokémon that have already been caught, simply go up to them, save, and get them into your party. Then, check to see if they are shiny. If they aren't, reset your game and do the process over again until they become shiny.
For legendaries, save when you go up to them and trigger the battle. When they appear in battle, see if they are shiny. If they aren't, reset your game and rebattle them until they become shiny.

Other shiny Pokémon you can more easily encounter are the Pokémon that are extremely common. Sure, you may be sick and tired of running into Zubats and Tentacools everywhere, but since you see them so often, your chances of finding a shiny one increases.
Try going into places where only one kind of Pokémon can be found, such as Whismur in Verdanturf Tunnel or Diglett in Diglett Cave if you are interested in shiny versions of those Pokémon.