Penelope is a teacher at a kindergarten in the Kalos region near Shoyo City. She appears to be at least 18 years old as she has a drivers license and access to a car.
She met
Ash Ketchum and his friends by coincidence while they were on their way to Shoyo City and her eyes met
Ash Ketchum and as per the Trainers rule, they had a Pokémon battle. She added the stipulation though that if she won the battle, he'd have to be with her for a while.
Serena mistook that as a romantic advance though.
Ash Ketchum lost the battle and he agreed to be with her for awhile. She took him to her school where he showcased his Pokémon to the children there.
Penelope and Nymphia go out to the forest or the meadows every day and bring back wild Pokémon they've befriended to show the children so they can interact with Pokémon.