In game, when a shiny Pokémon enters battle, stars will swirl around it. It will also have 3 stars on its summary screen. Besides from have slightly different colors, shiny Pokémon have no real advantage over normal Pokémon. The change in color is purely aesthetic.
In Gold/Silver, shiny Pokémon were introduced to most players with the infamous red Gyarados, which could be found terrorizing the citizens of Mahogany Town at the Lake of Rage. However, naturally shiny Pokémon occurring in the wild is a much less widespread occurrence. Players have a one in 8,192 chance of running into a shiny Pokémon randomly as they walk through tall grass, caves of other areas where wild Pokémon live. Shininess of Pokémon that hatch from eggs is determined at the moment you get the egg, not when it's hatched, with the same 1/8192 chance. Few players will encounter a shiny Pokémon more than once during an entire playthrough of a game, if ever. Shiny Pokémon are extremely rare and every effort should be made to catch them!