Flurry Heart is first indirectly mentioned in the season five episode The One Where Pinkie Pie Knows, in which Princess Cadance and Shining Armor announce to Twilight Sparkle and her friends that they're having a baby.
Depiction in the series
Flurry Heart makes her on-screen debut in the season six premiere, her reveal being witnessed by the Mane Six, Princess Celestia, and Princess Luna. Unlike Twilight and Cadance, Flurry Heart is a natural-born Alicorn; Celestia remarks that "the birth of an Alicorn is something Equestria has never seen".
To commemorate Flurry Heart's birth, the Crystal Empire holds a ceremony called a "Crystalling", in which she is presented to the public, and the light and love within the Crystal Ponies makes the Crystal Heart stronger. However, her crying causes the Crystal Heart to shatter, potentially ushering in an eternal winter. During Twilight and her friends' efforts to restore the Heart, Flurry Heart makes things difficult for them by flying all over, teleporting all over the place, and zapping things with her magic.
With the help of Sunburst, the ponies carry out Flurry Heart's Crystalling and restore the Crystal Heart. It is only after the incident is over that Flurry Heart is given her name.
As a natural-born Alicorn, Flurry Heart possesses more powerful magic than newborn unicorns. A blast of magic created by her sneeze alone can blast through several ceilings, and her crying shatters the Crystal Heart into pieces. In addition, she can perform multiple consecutive teleportation spells. After the Crystalling ceremony, her magic is more stable and controlled.
Role in the series
Flurry Heart appears in the series with the same role. Before her birth, Cadance appeared pregnant with her in a few episodes until the episode A Flurry Heart Attack, where Flurry Heart meets the Lodgers, and proves that she is a good fighter with her magic despite her age. She is even powerful and skilled beyond her years, being able to take on even all 3 of the Lodgers' opposing villain teams, humiliating even the powerful Professor Eagle Beak, and even tolerating absorbed magic and take on Tirek. She is even a quick learner, learning how to speak well, and even when her magic is balanced, she is still often haywire with them when excited.