Izayoi

is the human mother of the title character Inuyasha.

Manga series
Rumiko Takahashi states in Zusetsutaizan Ōgikaiden that Izayoi (or simply "Inuyasha's mother" as she has no official name in the original series) was the daughter of a wealthy lord, though the family had fallen into hard times. She also states that InuYasha was born in an isolated location and into poverty. She is always portrayed in fine clothing, however, with a solemn yet graceful appearance. Little is known about her life, including how she met the demon lord who would become Inuyasha's father, although it appears that the two shared a close bond - the sword, Tessaiga, was created to protect her, as was the Robe of the Fire Rat. In a rather unexpected turn of events, Sesshomaru's mother makes an appearance late in the series, bringing about even more unanswered questions. She remarks, however, how like her son Sesshōmaru is to his father in regard to his care of the human child Rin.

The character of Inuyasha's mother only appears once as herself in the series, and has no lines. The character is introduced by Sesshōmaru, Inuyasha's elder half-brother, who uses a yokai to make a perfect duplicate of Izayoi in order to trick Inuyasha into revealing the whereabouts of their father's demon sword, the Tessaiga. Sesshomaru blames Izayoi for his father's death, claiming that she had 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 to fill the gap. The ruse is eventually discovered, however, although not before the location of the sword is found.

Throughout the rest of the rest of the series, Inuyasha's mother makes no other appearances and Inuyasha himself does not reference her.

Anime series
Once again, her character remains known only as "Inuyasha's mother" and her introduction and character remains the same as in the manga series. The anime, however, adds a few extra scenes, showing a young Inuyasha in a lavish garden at a wealthy estate. He attempts to join the other nobles in a game of ball, but they toss the ball away and leave as he goes to retrieve he. InuYasha later recalls this was the first time he saw his mother's tears.

InuYasha the Movie: Swords of an Honorable Ruler
The third InuYasha movie provides us with details about Inuyasha's birth and how the Inu no Taishō died. Izayoi is the daughter of a wealthy feudal lord about to give birth to a child, an event that causes the samurai Takemaru no Setsuna to hold her captive in order to lure the child's father, Inu no Taishō, into a trap so he can kill the Daiyōkai. Though Izayoi told Takemaru to leave in order to spare him from the Inu Daiyōkai's wrath, Takemaru admits that he still has feelings of love for her, but that he will not allow her to give birth to a half-demon. He attempts to kill both Izayoi and the unborn child, but leaves before noticing she is able to give birth alive to Inuyasha before eventually dying.

The Inu no Taishō easily overpowers Takemaru's men and rushes to Izayoi's aide, only to discover his beloved dead. Fortunately, he is able to use Tenseiga to revive her. Inu no Taishou gives Izayoi the Robe of the Fire Rat and then orders her to escape with their son, whom he names "Inuyasha". She follows her beloved's instructions and leaves. Unfortunately, her lover was already injured badly in an earlier battle with the dragon yōkai, Ryūkotsusei. The injuries weakened the Inu no Taishou enough so that when the mansion collapses on both him and Takemaru, he dies along with Takemaru. This sets up Izayoi to raise Inuyasha on her own.

It is also shown in the series that InuYasha still tends his mother's grave, despite it being over 150 years since her death.

Inu no Taishō
Unfortunately very little is known about the relationship that Izayoi and the Inu no Taishō had. In Zusetsutaizan Ōgikaiden book, she is, however, listed as his "wife". Considering Sesshomaru's mother is still alive, there may have been a divorce, although polygamy (very common for Japanese nobility of the time) is more likely. Regardless, it is doubtful that both of Inu no Taishō's wives would live in the same household. There are numerous theories about how and when Inu no Taishō and Izayoi met, but it seems as though we will never know the truth. What can be said is that both appear deeply romantically affectionate towards one another. Izayoi refers to him as and the Inu no Taishō clearly has only her in mind after his battle with Ryūkotsusei: instead of tending to his wounds, he rushes to her side, only to find her dead and cradling their child (thankfully, he is able to use Tenseiga to revive her). Before he died he told her "Izayoi you must live a long life, live long and well with Inuyasha." It clearly shows he loved her.

Inuyasha
The half demon son of Izayoi and Inu no Taishō and younger half-brother Sesshōmaru. Inuyasha had a special relationship with his mother, whom raised her son alone until her death, and becomes very defensive when she is brought up. He prefers not to talk about her, which shows he misses her lot. According the the third movie, Inuyasha will occasionally visit his mother's grave to place new flowers. She loved and cared for her son wholeheartedly, and realized he would have a difficult life being half-demon.

Sesshōmaru
It is unknown if Izayoi cared for Sesshōmaru or not but it is known and seen that Sesshōmaru despised her a lot because she was human and caused his father to leave his mother.

Trivia

 * The character of Inuyasha's mother is not officially given a name in the manga or anime series and was variously known as simply "Inuyasha's Mother" or "Inumama". She is given the name Izayoi in the third movie and, although it may not be considered canon to the manga series or even the anime series, most fans will use the name Izayoi for simplicity.
 * Izayoi is the name given to the night of the 16th of every month. The Japanese believe that on that date, the night is more beautiful.
 * Rumiko Takahashi, the series creator, has a tendency to draw noble women in a very similar manner. Izayoi's design is not much different from other characters of high rank, apart from the color and design on her kimono.