Basics: Kishuu Arashi is the solemn, black-haired priestess from Ise Shrine. Arashi originally left Ise to help Hinoto, the blind dreamseer predicting the end of the world, and to seek out the Ten no Ryu, or the Dragons of Heaven, and recruit Kamui to their side. Arashi was helped with this "mission" of sorts by Daisuke (Hinoto's guardian and the cousin of Aoki Seichirrou, another Ten no Ryu).
Arashi is calm and well mannered, and generally does not appreciate excess attention. This is why when Sorata became infatuated with her, Arashi was less than thrilled. However, throughout the series we see Arashi open up to the other Ten no Ryu, forming almost a foster-family with them. She becomes a little more flustered and a little less.... violent about receiving attention.
Ise: Arashi is the "hidden maiden" of Ise Shrine. Within the X universe, Ise is a renown Shinto shrine of which Arashi plays the role of a priestess. For more information on how she came to be at this shrine, look at Arashi's past. In actuality, however, there are multiple Ise Shrines, which are probably the most renown form of Shinto shrines.
Shinto is an ancient Japanese religion which focuses on the belief of kami, sacred spirits which take the forms of natural elements (such as rain, wind, and fertility). The main goddess of the Shinto religion is Amaterasu, the sun goddess.
The Shinto religion puts great emphasis on purity and prayer. And although priests are allowed to be married, the miko, young ladies who assist them, must be pure and unmarried.
Arashi most likely falls under the role of a miko, as we know that her *cough* 'unmarried state' was very important for her role at Ise.
Ryu: Arashi was raised at Ise because it was known that she was a Ten no Ryu, a Dragon of Heaven, and was thus very important to the future of humanity. When she was sixteen, Arashi moved to Tokyo to fulfill this role. While she was waiting for everything to truly begin, Arashi helped Hinoto contact the various members of the Ten no Ryu.
Arashi has been "gifted" with the ability to conjure a sword from her hand. However, there is a requirement for this ability- that would be Arashi's purity. If Arashi were to lose her virginity, she would no longer be able to fulfill her role as a Ten no Ryu.
Arashi takes her role as a Seal very seriously. It is almost as though her entire life revolves around her destiny to fight on the side of the Seven Seals. Practically everything, prior to coming to Tokyo, came second to Arashi's role as a Ten no Ryu.
Past: Arashi's the daughter of a previous Ise priestess and a university student. When Arashi's mother became pregnant with her, she fled the shrine and thus Arashi grew up with only her mother to rely on. Unfortunately, her mother died when she was seven and Arashi was forced to live on her own, searching for food in the garbage. When Kaede, an Ise priestess, found her, Arashi was contemplating whether or not she should continue living under such conditions. Kaede took Arashi in and told her about her future and her destiny, and thus Arashi found reason to live.