Shiny Pokemon are extremely rare in the wild. When a Pokemon is 'shiny' its colouration is different. However, apart from that, the shiny Pokemon remains the same as any average Pokemon of its species.
But as I said, they are INCREDIBLY rare in the wild, so if you see one, make sure to catch it. You can also see if a Pokemon is shiny as it will be marked with a red star in the summary. Shiny Pokemon also make a sparkle sound when encountered, as well as having blue stars when sent out to battle.
There is an exception in Pokemon Mystery Dungeon: Gates to Infinity, where Ho-Oh can be encountered as a shiny Pokemon. This depends on pure luck though. Shiny Ho-Oh will be stronger than the regular one, and also have a different moveset. Sadly, this is only in PMND, Gates to Infinity.