The name of the family in charge of the greatest spiritual land in Japan. The word that's the representation of "trouble" in the world of magi.
An old lineage but not one of the famed ones (in fact, they're treated like heretics). Due to the "road" opened by one of their family heads generations ago (apparently the 3rd of 6 heads; also this was apparently from a secret method that he dug up), the head of the family has a high probability of inheriting "the use of magic".
This generation's contenders for succession were sisters, an ill-omen in any magus lineage, and at the end of a conflict for the rights of succession, the younger sister acquired the position of "magic-user".
The older sister crossed over to London and became renowned for her work, but she didn't get along and left soon. (More like, she was given a sealing designation.) Could have entered the highest part of the Association, but realized that she'd never be able to get out once she did.
The younger sister on the other hand took off while talking to "the clerk at the reception desk on the 1st floor lobby" because it was a pain in the ass.
During the 5th Fuyuki grail war, the older sister was in hiding while the younger sister was touring the country, without a care in the world. The little sister isn't good at anything other than destruction, but, the older sister is more of the skilled craftswoman type.
And a lot of people seem to be making this mistake, but to make it real clear:
T-o-u-k-o c-a-n n-o-t u-s-e m-a-g-i-c.
Generally, Association interest in the Aozaki family is mainly just based on them being successors of magic, rather than any sort of individual talent. Touko herself happened to be an exceptional genius with beautiful circuits that were average in number but overwhelmed all else through precision, a born set of mystic eyes, five senses that could feel the intricacies of the world, and knowledge adapting to the natural laws without wasting her own unique aspects. Even without the magic part, the Association would take notice of her genius, but the fact that she was supposed to be the inheritor of the 5th was more significant.
The Aozakis have been losing their circuits over the years and by the time of Aoko's parents, there had been practically none, dooming the magus history of the Aozakis to end in that generation, until Touko showed up, whose number of circuits and deviation was in the top of the history of the Aozaki's.
The Aozaki crest itself is probably related to the issue of "inheriting the Aozaki magic".
If she can't cast magic, what the hell did she do in the fourth movie?