The basic chance of a Pokémon being shiny, and why it changed in Gen 6.
Shiny odds are the chance that any single Pokémon you meet is the rare, alternate-colour version. Every encounter is its own roll of the dice.
Gen 6 did not change the shiny formula itself. It doubled the cut-off value that counts as shiny, which doubled the chance.
Shiny Pokémon first appeared in Gold and Silver. Red, Blue and Yellow have no shinies of their own, although some Pokémon moved forward from Gen 1 can turn out shiny in Gen 2 because of how their stats line up.
Many games add ways to improve the base rate: the Shiny Charm, the Masuda Method for breeding, and game-specific tricks like SOS chaining, mass outbreaks or Sparkling Power. See the other Odds and Methods articles for the exact numbers in each game. Odds are approximate and shown the way most sources quote them.