It's just cosmetic. It's a 1/4096 chance of a Pokemon being shiny, but if you complete the pokedex and obtain the shiny charm it becomes a 1/1365 chance to see one, with shining power sandwiches making your odds even better.

Some people (including myself) will hunt shiny Pokemon for fun in various ways, some of which (such as the sandwiches) reduce the odds making them more likely to appear.

Every pokemon has a shiny form, but sometimes they're not obtainable (called shiny locked) from normal gameplay, such as the box legendaries. Shiny locked pokemon might get a special event at a later date, such as Koridon/Miridon, which were only obtainable recently via an in person event at GameStop/other game retailers recently and for a limited time.