Shinies appear randomly, the "odds" of it, are rare, numbers do not matter because it is all a matter of luck and chance. The numbers "1/8612" or w/e really say that for every 8612 pokemon, you will find one shiny. Just keep attentive to details. At the beginning of every shiny pokemon battle, including ones against other trainers, a line of stars will appear, do loop a on the pokemon, and fly off the screen in a line. The pokemon will also be slightly discolored, seperating them against other pokemon ,which is the only difference. I will edit my post with pictures of examples for differences in colors between pokemon norms and pokemon shinies. Shinies may appear anywhere that pokemon do. Including legendaries.
Obvious differences:
(Normal)(Shiny)
(Normal)(Shiny)
(Normal)(Shiny)
Not so obvious differences:
(Normal)(Shiny)
(Normal)(Shiny)
(Normal)(Shiny)
Those are practically the differences of shinies though. You get a different colored pokemon and 'bragging' rights, if you have friends that like pokemon. Welcome to the Shiny Hunters Club. If you have any questions, feel free to ask. Remember, there aren't stupid questions, just other people who think they are.