Appearance: In his true form, Inutaishou was a gargantuan, pure white dog. His other forms, though he certainly was capable of shapeshifting, are unknown.
Appellations: Inutaishou, Inutaishou-sama, oyaji (by Inuyasha), chichiue (by Sesshoumaru). When talking to Inuyasha, Myouga refers to Inutaishou as "chichi-gimi", an archaic (even in feudal Japan!) and very respectful term for "father". Similarly, Toutousai uses "oyaji-dono", in which "oyaji" is a rude word for "father" and -dono is an old-fashioned, rather respectful name suffix.
Meaning of Name: "Inutaishou" translates to "chief/leader of dogs". It is by no means certain that it is the true name of Inuyasha and Sesshoumaru's father, but it is the only one known.
Impressions: Inutaishou is father to both Inuyasha and Sesshoumaru, a dog taiyoukai (great demon) who ruled the Western Lands, assumed to be the southwestern island of Kyuushuu. To whom he sired Sesshoumaru is not known. He fell in love with a mortal woman and took her as his mate -- the offspring of this union was half-blooded Inuyasha. His death is also wreathed in mystery, but he definitely had plans for his feuding sons. Toutousai, a master swordsmith, fashioned two swords, the Tetsusaiga and the Tenseiga, of Inutaishou's fang. They were left as heirlooms for Inuyasha and Sesshoumaru, respectively, and Inutaishou hid his grave inside Inuyasha's eye to ensure that the Tetsusaiga would end up in the hands of its rightful owner!

Nothing definite can be said about which son was intended to succeed Inutaishou as Lord of the Western Lands, nor indeed much about his personality. Humans were in his favour, especially since he took one to mate, and the anime suggests (in a filler story arc) that he was a just and fair ruler, defending his lands and those who sought his protection.
First Appearance: manga act 15, anime episode 6 (not alive, though)

Appearance: The sole time she has been seen in the story, Inuyasha's mother was dressed like a lady of the court. She had sad dark eyes and black hair that reached almost to her ankles. She wore the heavy, multi-layered, colourful kimono and dark hakama of a Heian era noblewoman, a style of dress distinctive to the ladies of the imperial court.
Appellations: ofukurou (by Inuyasha, a colloquial word for "mother"). Myouga refers to her by "haha-gimi", a very old, obscure and respectful term for "mother", when speaking to Inuyasha.
Impression: The name of Inuyasha's mother is not known. She fell in love with a taiyoukai, Inutaishou, and became his mate. It is likely that she was of noble birth, a conception suggested by her rich garments, but then again she was the mate of a youkai lord so that certainly might account for her appearance. She gave birth to a hanyou son, Inuyasha, but she died when Inuyasha was still very young. She was a gentle and loving mother, always trying to protect her son from the taunts of the people of the court (who were youkai, of course). One noteworthy thing is that the Tetsusaiga was forged because of Inutaishou's love for and desire to protect her; no full-blooded demon may touch the blade without serious injury, which speaks in favour of humans and their half-blooded son.
First Appearance: manga act 12, anime episode 5 (in ghost/memory form)
Seiyuu (Japanese): Inoue Kikuko
Voice Actor (English): Alaina Burnett