It is an indicator of a Pokemon met in a unique manner, typically through an event.
A fateful encounter is a flag attached to certain event Pokémon and some wild Mythical Pokémon that indicates they were met in a special manner.
In Generation III, it is used as an anti-cheat measure for Mew and Deoxys, penalizing players who obtained them through unofficial means. In Generation IV and V, starting with Pokémon Platinum, the presence of this flag has been used to trigger special events if the player has a certain Pokémon that was met in a "fateful encounter".
Pokémon obtained via event distributions often have the fateful encounter flag set. Events distributed prior to Pokémon FireRed and LeafGreen did not use this flag, as it was not used in Pokémon Ruby and Sapphire.
Fateful encounter (Bulbapedia)
