Shiny pokemon are incredibly rare, due to the chance making sure only a few appear in game. The rarest right now is judged to be mewtwo, with a chance that you may not notice the differences between a normal or regular mewtwo (Only differences being the tail and the eyes), with it also being a legendary, another thing which depends on chance to appear.
Shiny pokemon have a different colouration than regular pokemon. This changes between different pokemon however, as different species of pokemon have different shiny changes. An example of this is Lapras and Poliwrath, 2 predominantly blue pokemon. Lapras turns a light shade of purple when shiny and Poliwrath turns into a lighter blue. It may even become a small difference, such as lunatone having only the colour of the eye changed, red to blue.