Your first question is probably what a "shiny" even is. A Shiny is a special type of Pokémon. Most prominent is its coloration, which is generally quite different from a normal Pokémon’s. For example, a normal Pikachu is yellow; a Shiny Pikachu is more orange-gold in coloration.
Besides the different color, there is no actual difference between a Shiny Pokémon and a normal one. However they’re quite prized for their rarity (and often appearance) - in Pokémon Black/White 2, a rare medal was even awarded for finding one.
In some games, you were guaranteed to encounter a Shiny Pokémon on certain occasions - for example, the Shiny Gyarados at the Lake of Rage in Gold/Silver/Crystal, or the Shiny Haxorus in the Nature Preserve of Black/White 2.
Sometimes Nintendo even distributes special Shiny Pokémon, like the Shiny Pichu for the Spiky-Eared Pichu event in HeartGold/SoulSilver.
The odds of finding a Shiny Pokémon in the wild are pretty low. Reportedly, the odds are 16 in 65,536, which is equivalent to 1 in 4096 or 0.0244% . So, yeah, they’re rare, although twice as common as in previous generations, for what it’s worth…
In some generations, there are methods to boosting the rate of finding Shinies. Most notably, you can complete the National Pokédex for the Shiny Charm , which triples the Shiny-finding rate. It’s not a lot, but, heck, 3 for 1,000 hours of gameplay is significant enough for some.
Less obvious methods exist too, which are ironically more effective. There’s the Matsuda method (see the Breeding section for the details), where you breed with a Pokémon from a foreign country to greatly raise the chance of hatching a Shiny - with the Wonder Trade function in X/Y, that becomes very significant!
Best of all, there’s PokéRadar chaining , which can make Shinies 40 times more common by simply finding the same Pokémon again and again through the PokéRadar. X and Y has also added the similar method of Chain Fishing . It’s a bit of effort, but like they say: "No pain, no gain."