Shiny refers to a rare body color of Pokemon. The fastest way to get them is to buy Shiny Pokemon from PKMBuy. In Pokemon, Pokemon have more than one in eight thousand chances that they will appear different from ordinary Pokemon when they are spawned. Each type of Pokemon only corresponds to one shiny color.
Shinies are found by simply trying to catch Pokemon and seeing if they display as an alternate color in the battle and post-battle screens. They won't appear as a different color in the field, so you have to attempt to catch them first.
Another thing to note is that the regional form of some Pokemon may also be shiny. Just like standard Pokémon, if there is a shiny icon above the CP, you can judge whether they are shiny. PKMBuy is a professional Pokemon trading store, you can Buy Shiny Pokemon on it.