Battle/Breed a Pokemon. In X & Y, there is a 1/4096 chance of getting a shiny Pokemon without doing any special methods.
Battle a horde of wild Pokemon. The chance of getting a shiny Pokemon in a horde battle is multiplied by 5 to about 1/819 (1/819.2).
Chain using the PokeRadar. Using the PokeRadar and getting a chain of 40 or more increases your chances up to 1/200. Here is an excellently written guide by Smogon.
- Alternatively, you can also go consecutive fishing, and get a chain of 40+ to increase your chances up to 1/200. Here is a guide right here on the site.
Use the Masuda Method by breeding two Pokemon of different countries to increase the chance of getting a shiny Pokemon. The current chance is unknown.
Complete the national Pokedex (see all of the Pokemons) and get a Shiny Charm, which increases the chances of finding shiny Pokemon. The current chance is unknown.
Here is a similar question.