Izayoi

Izayoi is the mother of the title character InuYasha.

History
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. 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, with the regret that her son would be discriminated by both humans and demons for his human heritage. She died when InuYasha was very young.

InuYasha the Movie 3 - Swords of an Honorable Ruler
Izayoi appears at the very beginning of the movie in a flashback. Izayoi was the daughter of a feudal lord, and at one point was held captive by the samurai Takemaru. Takemaru planned to use her to lure Inu no Taishou into trap to kill the demon lord. Though Izayoi told Takemaru to leave in order to spare him from the dog demon's wrath, Takemaru said he loved her and killed her, but not before she gave birth to InuYasha. The dog demon overpowered Takemaru's men only to discover his beloved dead, but then he used Tenseiga to revive her. Inu no Taishou gives Izayoi the Robe of the Fire Rat and then orders to escape with their son, whom he names InuYasha. She follows the dog demon's instructions and leaves, unfortunately her lover was already injured badly in a earlier battle with the dragon demon Ryukotsusei. The injuries weakened the Inu no Taishou enough to when palace collapses on him and Takemaru, that he dies along with Takemaru. This sets up Izayoi to raise InuYasha on her own.

InuYasha's mother's name is never given in the manga, or the anime series. She is given the name Izayoi, in the third movie. Sesshomaru, InuYasha's elder brother, uses a yokai to make a perfect duplicate of Izayoi in order to trick InuYasha into revealing the whereabouts of their father's sword, the Tetsusaiga. Sesshomaru blames Izayoi for his father's death, claiming that she made the Inu no Taishou weak which lead to his death. The manga never elaborates on Sesshomaru's claim, however the third InuYasha movie attempts fill the gap. The Inu no Taishou went to rescue Izayoi right after his battle with Ryukotsusei despite having life threatening injuries. The wounds ultimately weaken him enough to where he is killed by a collapsing palace, something he most likely would have survived had he not been injured. Since the events of the third film are never mentioned in the manga, they are not exactly canon. Sesshomaru could have also meant that had the Inu no Taishou never loved Izayoi, he would have been strong enough to kill Ryukotsusei.