A Shiny Pokémon is a Pokémon with an alternate color palette, determined randomly at encounter or hatching, at a base rate of 1 in 8192 in Generations II to V and 1 in 4096 from Generation VI onward. Shininess changes appearance only, never stats, in the modern games.
Shiny Pokémon debuted in Generation II, where shininess was calculated from a Pokémon's IVs, so a Shiny was also guaranteed above average stats for its time. From Generation III onward shininess is decided by a hidden value comparison between your trainer IDs and the Pokémon's personality value, fully separate from stats. The famous red Gyarados at the Lake of Rage in Gold and Silver is a scripted Shiny, placed there so every player would meet the mechanic once.
The base odds, often called full odds, are 1 in 8192 from Generation II through Generation V, and 1 in 4096 from Generation VI onward. Hunting at full odds means using no boost at all, and dedicated hunters count encounters into the thousands, sometimes passing the odds several times over before anything shines. That variance is the point of the hobby for many of them.
Three boosts matter most. The Masuda method, breeding two Pokémon from games of different languages, gives extra rolls: about 1 in 1639 in Generation IV, 1 in 1366 in Generation V, and 1 in 683 from Generation VI onward. The Shiny Charm, introduced in Pokémon Black 2 and White 2, adds two extra rolls for most encounters, taking Generation VI onward wild odds to about 1 in 1366. Masuda breeding with the Shiny Charm from Generation VI onward reaches about 1 in 512, which is why most modern Shiny hunters breed rather than wander. Method specific boosts such as outbreak bonuses in Generation IX stack on top of these basics.
In play"I am 4,000 encounters into a full odds hunt and still have nothing to show for it."