From Final Fantasy IV, she's at first a young caller (revised as "summoner" in later translations)
girl whom you meet in a village after you and your friend accidentally destroy it in a fire along with her dragon mother in the
cave that you went through to get there. Rydia totally hates you for doing that, but eventually learns to forgive you. She then joins your
party and becomes a pretty okay spellcaster.
Later on, she gets swallowed by Leviathan and is sent to the Land of Monsters. The
flow of time is accelerated there, allowing her to rapidly grow and quickly become an adult. However, she also loses all of her powers as a white mage, but at the same time learns a bunch of awesome black
magic spells and summons.
From then on she returns to your
party and becomes one of the most kickass characters in Final Fantasy
history, along with being a combination black mage/summoner and having really nice-looking green hair. You can't beat that, folks.