A Breeder is a monster owner who raises monsters for all sorts of purposes. If he or she wishes to raise them to compete in Monster Battles, a Breeder must find a Monster Association to register with (usually the one sponsored by their country or kingdom).
In most games, there is a dialogue box where you can check your Breeder Data, or all of the information about your character. This may include your age in game years, your experience level based on the training your monsters have received (called Rank), and your overall win/loss record for battling. It will often also tell you how much money you have won in the course of your career.
Later games use a points system called Breeder Points to quantitatively measure your progress.