Synchronize is an ability introduced in Generation III. So far, 4 Pokémon have this ability.
When the Pokémon is burned, paralyzed, or poisoned by another Pokémon, that Pokémon will be inflicted with the same status condition. Synchronize will have no effect if the opponent has a type, Ability, or team condition (e.g. Safeguard) that prevents it from being burned, paralyzed, or poisoned. However, Synchronize does affect an opponent behind a substitute.
Outside of battle
If the bearer is leading the party (even if fainted), any wild Pokémon encountered has a 50% chance of having the same Nature as the bearer. This even affects encounters with certain Legendary Pokemon, but does not affect "Gift" Pokemon (Pokemon not caught via battle) such as Mystery Gift Pokemon, the Eevee recieved in Venesi City, etc.
Pokémon with Synchronize