The Hyuga is one of the four noble clans of Konoha, along with Akimichi, Aburame, & Uchiha clans. The symbol of this clan is the Bagua, which is represented by the yin-yang surrounded by the 8 trigrams. Every member of the clan is born with a kekkei genkai doujutsu known as the Byakugan that gives them extended fields of vision and the ability to see through solid objects and even the chakra circulatory system, amongst other things, and expel chakra from any of the tenketsu in their body.
The clan is split into two sections, the main and side houses, which creates a system that helps to protect the the Byakugan from someone who would try to take it. The main branch runs the clan while the branch family protects them, sometimes even with their life. To keep from having their eyes stolen, The members of the Branch family have a seal placed on that won't disappear till death.
In the past, there have been several attempts on the part of Kumogakure to acquire the dōjutsu of this clan. This event culminated in the death of the would-be-thief: the Head Ninja of Kumogakure and the demands of Hiashi's corpse who had killed him according to the stipulations in the peace treaty that the two villages had signed. Ultimately, Hizashi was the one to sacrifice his life for the village and clan, sealing away his Byakugan as he died and avoiding war. This event would become known as the Hyūga Affair.
Ao of Kirigakure was successful in obtaining one Byakugan from a hyuga clan member he fought some time in the past.