Izayoi is the mother of Inuyasha. In the manga, she has no name.
Very little is known about Izayoi as she died before the events of the series takes place and because Inuyasha rarely speaks about her, most likely because he is still saddened by her death. She was the daughter of a feudal lord and at some point she met and fell in love with Inu no Taishou, who returned her affection. Although the two form a relationship it is never specifically stated if the two were married or officially mated. When Inu no Taishou died she was forced to raise their son Inuyasha alone. She was sad that Inuyasha would be discriminated by both humans and demons for his hanyou heritage. It is unknown when she died or what caused her death, but she died when Inuyasha was fairly young.
Inyasha's mother's name is never given in the manga but she is given one in the third movie of the series. Sesshomaru blames Izayoi for his father's death, claiming that she made the Inu no Taishou weak which lead to his death.