Breeders are a prestigious group within the world of Seikoku no Dragonar.
Breeders are those who, after presenting themselves to the Mother Dragon, were chosen by her and implanted with a baby dragon. The Breeder is marked with a stigmata called a Seikoku, which contains the infant dragon until it is time for it to manifest outside of its pal's body. Thereafter, the Seikoku acts as a link between the two.
Breeders may eventually achieve the rank of Dragonar by helping their Pal become a Maestro, or Holy Dragon.
If a Breeder desires, they may renounce their status, and pass their Seikoku, and the Dragon it contains, to another person, making them a Breeder. This is apparently very damaging as it destroys the body part possessing the Seikoku of the original Breeder. Ash Blake is the only one known to have done so, and it destroyed his arm, and nearly killed him. Something that's also important to remember is that Ash did this during the process of having the Seikoku branded upon his body, it's quite likely that he wouldn't have been able to give it up if said process had finished.