Original Trainer
The Original Trainer (OT) of a Pokémon is the trainer who caught or hatched the Pokémon in question first. The original trainer's name is used in conjunction with their ID number to verify which Pokémon are outsider Pokémon by the games.
In Pokémon Colosseum and Pokémon XD: Gale of Darkness, Shadow Pokémon that are snagged will consider the snagging trainer to be their OT.
In games from Generation III onwards, the color of the OT's name will differ on most status screens, with it being blue at some times and red at others. This means nothing more than the gender of the OT, with blue meaning that the OT is male and red meaning the OT is female.
Some Special OTs
Sometimes, there are special OT names assigned to Pokémon if they come from a non-standard Pokémon game or are received as a prize. See List of notable ID numbers for examples.