The main ones are four human teenagers and their fairy partners (called "muglox" according to ShoPro/Viz Media materials[1]).
In addition, there is a large cast of supporting characters, including regular characters, some of whom change with each arc (original, Wonderful, Golden, and Enchanted seasons, as well from the Murumoya fansite; for details, see List of Mirmo!
One day, she was walking around in the park with her dog who finds Yatch's mug.
Before that, Yatch ran away from Azumi's home, taking his mug with him, but later he got trapped in it by accident.
Rie followed the instructions that were written under Yatch's mug, so Yatch appears and sees Rie as a very wonderful, organized, and hard working person (compared to Azumi, who is the exact opposite.)