Shiny Pokemon are somewhat akin to gold dust. Perhaps you've heard of them, maybe your friends have some or, who knows, you could be swimming in them. In any case, they are generally very rare and valuable Pokemon to have.
As their name suggests, Shiny Pokemon "shine" when they enter battle. They also have a red star next to their markings in the summary screen. However the easiest way to identify a Shiny Pokemon is usually their different colour compared to a regular Pokemon.
We'll describe a very common and iconic example: a regular Magikarp is red, but a Shiny Magikarp is gold. Shiny Pokemon will stay shiny when they evolve, but their shiny colour can change. Using our previous example, a shiny Gyarados is red (and not gold) instead of blue.
Apart from the rarity and the unique characteristics mentioned above, there is no difference between a normal Pokemon and a Shiny Pokemon. So, in terms of stats, a Shiny Pokemon is no better or worse. The main reason to have a shiny is for bragging rights or if you like the shiny colour.
How to Find
So you know about Shiny Pokemon, but how do you find them? Well, just like normal Pokemon: naturally in the wild or by hatching Pokemon Eggs. However, under normal circumstances, there is a 1 in 4096 chance of a Pokemon being shiny.
If you've never encountered a shiny ever, that's the reason why. On average, you can probably expect to encounter, say, 500 wild Pokemon during a playthrough. But that's not nearly enough to guarantee a decent chance of finding a Shiny Pokemon.
Fortunately, there are a few methods to literally improve your chances.
Shiny Charm : After completing the Alola Pokedex, you can receive this key item from the Game Director working at the Game Freak offices in Heahea City. Simply having it will improve your chances of finding (or hatching) Shiny Pokemon. By completion, we mean you gotta actually catch them all! Fortunately, this doesn't include the National Pokedex, only the Alolan one.
Masuda Method : Named after the Pokemon director who shed light on the existence of this method. If you breed using a Pokemon from a different language version, the chances of hatching Shiny Pokemon is drastically increased. You can tell if it's from a different region by looking in the Summary of the Pokemon, the region abbreviation will be in the top right.
Ally Chaining : When a wild Pokemon is low on HP and afraid, it may call an ally Pokemon (see S.O.S. Battles ). This can continue to occur so long as there's only one wild Pokemon present. The more ally Pokemon summoned, the greater the chance the next ally Pokemon will be shiny.
The following Pokemon, when obtained normally in-game, can never be shiny: 205, 285 to 302. However they may be shiny if obtained from Mystery Gift.
A caveat: Mystery Gift Pokemon can only be shiny if they are supposed to be shiny in the first place. If they aren't supposed to be shiny, they will never be shiny.