Shiny Pokemon or 'Shinies' are categorized by their unusual colour. Often they have colours that are muted or that belong to a completely different colour palette than their normal conterparts. Shiny Pokemon are also gifted with higher stats than their normal forms and it is this, combined with their unusual colouring, that makes them so popular with Pokemon trainers.
Probably one of the most well known Shiny Pokemon in the Pokemon Universe is the Red Gyarados, which appeared in the Pokemon games Gold, Silver & Crystal in the Lake of Rage as well as making a cameo in the Pokemon anime. The second well known Shiny Pokemon in the Animeverse is Ash's/Satoshi's Noctowl which is a soft gold colour rather than the species usual brown. When exiting their Pokeballs in the games and anime Shiny Pokemon really do shine. This is normally represented by stars or a bright flash of light.
In Pokemon Battle Arena, shiny pokemon are given a 25% HP boost, therefore, pokemon are sold for up to 10X more money.