Caution: spoilers ahead for The Witcher
The Witcher season 2 explains the source of Ciri's magic is the Elder blood running through her veins - what does this mean? Magic is hardly rare in the world of The Witcher, with everyone from Kaer Morhen to Aretuza able to cast a spell or two. Nevertheless, Princess Ciri's magical gifts have amazed and terrified the few to actually witness them. In The Witcher season 1's past timeline, Geralt watched as Ciri's mother exploded into a magical tornado - an emotional outburst after her beloved Duny almost perished. Ciri proved her inheritance of that power some years later during Nilfgaard's invasion of Cintra, escaping capture by unleashing a similarly powerful and uncontrollable force.
Neither Geralt, nor Ciri herself, could pin down exactly where her power came from. The Witcher's magical characters typically draw spell-casting energy from Chaos, allowing them to cast illusions, ignite flames, rend the Earth, and ransack the mind. Geralt admits that the power he's seen Ciri and her mother wield bears little resemblance to any Chaos he's experienced thus far across long decades of existence, with the young Princess' outbursts considerably more destructive than your typical untrained mage. When Ciri begins her Kaer Morhen training in The Witcher season 2, however, the truth is revealed.
Vesemir discovers the rare flower Feainnewedd growing where Ciri's blood spilled during training. According to Triss, this indicates Ciri is a carrier of Elder blood - something believed to have died out generations prior, and as such, isn't fully understood. In The Witcher's wider lore, ancient elf sages engineered a mutated gene, hoping to create a Child of Elder Blood who would exceed them in power and open a path between worlds. The last of those to inherit Elder blood was Lara Dorren (the female elf that Ciri and Triss encounter in The Witcher season 2, episode 5's Valley of the Soul sequence). Lara's is the very same tale Nivellan tells Ciri in The Witcher season 2's premiere. She bred with a human mage, and the line was believed to have died out thereafter.
Instead, a latent Elder blood gene continued through Lara's descendants, which include Ciri's grandmother. When that DNA became "activated" by breeding with another non-magic gene carrier, the Elder blood was reborn once more in the female inheritors. This explains why Queen Calanthe mentioned her own mother possessing the same power as Pavetta in The Witcher season 1. It skipped Calanthe herself, awakened in Pavetta, then emerged fully in Ciri as the prophetic Child of Elder Blood. Unfortunately for her, that makes Ciri a Source - a wellspring of great untapped magical power that threatens to bubble over if left unattended, or during stressful situations (like watching your entire family and kingdom die).
By design, Elder blood as a source for Chaos is far more powerful than any regular magic seen in The Witcher, which explains why Pavetta and Ciri's powers are so unfamiliar to Geralt. The source of her magic also accounts for the terrified reaction of Triss, and others who perceive Ciri's true nature. The Child of Elder Blood was intended as an avenger for the elves, who would then lead them toward salvation, but the tragic story of Lara Dorren means the prophecy has become a portent of doom for all races. If Ciri could control her magical abilities properly, she could wield power on par with the Conjunction itself.
As explained by Vesemir, Elder blood was a rumored ingredient in the original birth of the Witchers. The mutagen used to strengthen Geralt and his brothers was destroyed long ago, but with a new source of Elder blood to hand, more Witchers can be made. While Ciri is a godsend for Kaer Morhen, others will view her solely as a bringer of death and destruction - or else a weapon to be wielded on the battlefield. Elves, on the other hand, may view her as a long-awaited hero or a genetic human aberration. With The Witcher season 3 already confirmed, the Netflix adaptation will surely dig into Ciri's powers in future.