How can you increase the shiny rate?
From serebii.net, if you breed a pokemon from another region different to your game with another pokemon, you will have a higher chance of getting a shiny pokemon. Colloquially called the Masuda Method after the developer Junichi Masuda, this can increase your chance of breeding a shiny from 1 in 8,192 to 1 in 2,048 (Generation 4) or 1,365.3 (Generation 5). Generation 6 figures are unknown at the moment, but to be honest I've heard a lot of people getting shinies so it could have been lowered even more.
As noted by Nolonar it is required that both parents must come from a game with different language. For instance, if your game is in English and both parents are Japanese, the Masuda method will not work.
Without any method, the chance of encountering, obtaining, or breeding a shiny pokemon is 1 in 8,192.
Although this doesn't specifically increase your shiny rate, there's an item called a PokeRadar which can use a method called chaining to find shiny pokemon in the wild. A good guide I found is here
In addition to the other region pokemon, Poke Radar, and "just breed piles of stuff" methods, there is a new Consecutive Fishing method which appears to significantly increase Shiny occurances but exists only in Generation 6/ Pokemon XY.
It turns out that if you manage to get a chain of consecuctive fishing hooks, it will gradually increase the likelihood of finding a Shiny Pokémon, with the exact figures still to come. The higher the amount of Pokémon hooked, the more likely you get a shiny Pokémon, but as with PokéRadar, this is not a guarantee.
There are a few things that will break this consecutive streak, however. If you move from the spot, the streak is broken. If you pull in the rod too early, the streak is broken. If you don't get a nibble, the streak is broken. With the latter being a possibility, it's logical to have a Pokémon with the ability Suction Cups as your lead in order to increase the likelihood of hooking a Pokémon.
Some more info on fishing streaks is here:
What we know for sure breaks a streak:
-Getting the "Nothing seems to be biting..." message (Random, but the odds are manipulable)
-Getting the "No! You reeled it in too fast!" message (Pressed A too soon)
-Getting the "No! You reeled it in too slow!" message (Pressed A too late)
-Exiting the area in which you're fishing
-Getting into a battle that isn't through fishing
-Exiting the game
The shiny rate seems to increase significantly by this method and reports seem to indicate around 150-200 consecutive catches may grant a shiny. The problem of course is you can only catch shinies that can be fished up in that area, and it's mostly luck as to which pokemon you get (unless you want a shiny lucdisc or Magikarp, in which case use the Old Rod).
200 catches may seem high, but it's miles ahead of the one in 8000+ chance normally and still significantly better than the 1 in 1,365.3 chance the gen 5 foreign Ditto chance had (it is unknown if the foreign ditto method has improved in gen 6 yet).
In generation II, having a shiny parent did raise the chances of hatching a shiny pokemon.
However, after gen II, the chances of a shiny are NOT increased if either of the parents of shiny. So if you're playing anything other than Gold, Crystal, or Silver, having a shiny parent won't increase the shiny rate.