Root is a character appearing only in the 2006 film based on the first four Earthsea books. He doesn't appear in any of Ursula Le Guin's stories.
In the film, he appears as a minister and senior advisor to the older Prince Of Enlad (Arren's father). He is an old man, encompassing years and years of wisdom. He reveals the story of Dragons and Humans' brotherhood as one.
He is a trusted elder and help of the Prince and is well-versed in the talk of the Balance.