Kagome's Family
Sota Higurashi

Sota is the younger brother of Kagome, and is a third grade student. He is jealous of her adventures in the Sengoku Jidai, but Kagome claims it is too dangerous for him to accompany her. He once tried to go through the well but had no success.
Mama

Little is known about Mrs. Higurashi's past; her first name is not even mentioned. She is portrayed as a cheerful, pleasant woman who seems quite content to keep house for her children and their grandfather; to date, it has been clarified in a separate novel that Grandpa is the father of her husband. She appears to be very permissive about Kagome's life, and does not seem to mind at all that her daughter spends long periods of time in the distant past.
In episode 69 of the anime, "The Terror of the Man with No Face", Kagome's brother Sota bemoans the fact that only Kagome can go to the feudal era, and Kagome's grandfather comments that it is her fate. Kagome's mother makes a soft noise of apparent agreement. This is the only real evidence given by the series or movies to date that her mother understands what's going on, and while being concerned (like any normal parent would be), Mrs. Higurashi simply accepts that this is her daughter's karma and does what she can to make her daughter's adventures in the Sengoku Jidai easier. Since the Higurashi family is portrayed from the beginning of the series as being Shinto, as well as the guardians of several items with rich mythological/spiritual histories (the Sacred Tree, the Bone Eater's Well, and So'unga), Mrs. Higurashi's acceptance of karma without apparent struggle or rancor is not particularly surprising. Cultural differences between Japanese and American society seem to be the source of misunderstandings about Mrs. Higurashi's attitude in the fandom.
Mrs. Higurashi's husband and the father of Kagome and Sota is never shown or seen in the series. But in Shousetsu InuYasha, a novelization of the first eleven chapters, a side note says Papa died a long time ago because of an accident, and Kagome, along with her mother and younger brother Souta moved into Papa's parents' home, which was the residence on the shrine property. Jii-chan, who was the shrine's caretaker, was alone after his wife passed away. Mrs. Higurashi has only spoken about him to Kagome once in the series to date:
In episode 48, "Return To the Place Where We First Met", Kagome stands beneath the Sacred Tree, where she first met InuYasha in the Sengoku Jidai, realizes that she loves InuYasha, and falls to weeping. Mrs. Higurashi comes upon her crying daughter and comforts her with the story of how her husband proposed to her beneath this very tree, and that being in the presence of the tree gave her the clarity of mind to think past her initial bewilderment and accept his proposal.
Mrs. Higurashi is not only accepting of Kagome's karma, but also of InuYasha. Since InuYasha is the only other character in the series who can travel through the Bone-Eater's Well and into the modern era, he has often interacted with Kagome's family and friends on occasions when he comes through the Well to fetch Kagome away from her modern obligations and back to the quest for the Shikon no Tama in the Sengoku Jidai. Mrs. Higurashi has always attempted to make InuYasha feel welcome, apparently unfazed by his appearance or his paternal heritage, and has no qualms about entertaining him at the house while Kagome is at school or letting him spend time with Sota. This may be because she recognizes that he is the half-demon from the ancient legend about the Sacred Tree (as told by Kagome's grandfather in InuYasha the Movie: Affections Touching Across Time) and therefore an integral part of Kagome's karma. Another possible explanation is that she recognizes that InuYasha is highly protective of Kagome and much stronger than an ordinary human due to his demonic paternity, and therefore in a unique position to keep her daughter safe from the unpredictable dangers of the Feudal Era.
Grandpa

Grandpa Higurashi, or simply Grandpa (or Jii-chan) is the father of Kagome's father in the popular anime and manga series, InuYasha.
He sees himself as the guardian of the family shrine. He tries to sell trinkets (such as fake Shikon No Tama jewels and ofudas) and goes through the motions of being a priest without any obvious spiritual powers. He also spends his days thinking up complex illnesses to excuse Kagome from the school she misses while in the Feudal Era.
With a comical touch, he often grieves over why his grandchildren Kagome and Sota take his practices of Shinto lightly, and that they are not interested in history of the sacred shrine at all. He is very dedicated to the shrine; in InuYasha the Movie: Affections Touching Across Time, he is delighted that the sacred tree in the shrine grounds has blossomed for the first time in many years and is devastated to see the snow (caused by the time compression of the past) cause the blossoms to fall too soon, resulting in Sota helping him in a 'chant' to bring back the blossoms. Much to his surprise,it didn't work.
Grandpa does possess some amount of knowledge about demons and exorcism. However, at an advanced age and in a time where there are no demons to cast out, his spiritual powers are minimal. When he does exorcise demons, they're usually false "demons" or his exorcisms don't work. He has tried to exorcise InuYasha a few times because of his doglike ears though. Despite his weakness he did succeed once with a demon. When the Flesh-Eating Noh Mask appeared the charms he placed on his face and mouth prevent him from being consumed by it.
He has a family heirloom consisting of a jar with a strange looking root in green liquid (which looks a bit like a ginseng) that can cure any illness. This is broken accidentally by Sota in episode 89 of the anime.