The Shiny Charm increases the chance of encountering a Shiny Pokémon, both in the wild and through breeding. This is done by generating two extra personality values to attempt to find one that is Shiny, effectively resulting in a tripled chance to find Shiny Pokémon compared to the base chance; from 1/8192 to approximately 3/8192 (roughly 1/2731). This stacks with most other modifiers which increase the Shiny encounter rate, such as the Masuda method.
Generation VI onward
In Generation VI, the base encounter rate for Shiny Pokémon was increased from 1/8192 to 1/4096. As such, the Shiny Charm now increases the chance of encountering a Shiny Pokémon from 1/4096 to approximately 3/4096 (roughly 1/1365).