Shiny Pokémon are Pokémon that have an unnatural coloration. The coloration can be mild - only a few shades darker or lighter than the original - or drastic. Pokémon have a very rare (1/8192 by default) chance of being shiny, and shiny Pokémon are much sought after for their unique look and rarity. A shiny coloration gives no advantage in battle, however. They are not to be confused with Boss Pokémon. A shiny Pokémon will emit particles and have a star under its level while in battle with a trainer. Wild shiny Pokémon will not engage in battle with other wild Pokémon.
Pictures of the shiny forms of each Pokémon can be found here.