A breeder is a monster owner who raises monsters for all sorts of purposes. If he or she wishes to raise them for sport, 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 as a breeder based on the training your monsters have recieved, and your overall win/loss record for battling. It will often also tell you how many golds you have.