Below are some of the characters that Kagura relates with on a regular basis, followed by a brief summary of their relationship. These summaries are mainly based on the anime, but I've also included how some of the relationships develop further in the manga. Obviously, these contain major spoilers for both the show and manga.
Name: Naraku
Relationship: "father"/creator
Summary: Kagura is one of Naraku's incarnations, but she harbors a deep hatred and resentment for him. She despises that he uses her as a pawn, with no concern for her safety or well-being. As a result, Kagura is constantly betraying him, often bargaining with his enemies in the hopes that they will destroy him. She is unable to kill him herself as he literally holds her heart in his hand.
In the manga Naraku eventually grows tired of Kagura's betrayal. When he feels she is no longer useful to him he grants Kagura her freedom, only to poison her moments later.
Name: Kanna
Relationship: "sister"
Summary: Though she appears younger, Kanna is actually the older "sister" of Kagura. She is completely void of emotion and is unquestionably loyal to Naraku. Kagura is unable to trust Kanna; she often reveals Kagura's deceit to Naraku with her mirror. Kanna is undoubtably Naraku's "favorite". He takes much better care of her, often keeping her out of harms way while sending Kagura on reckless and dangerous missions. As a result of this Kagura resents her "sister" as well.
Name: Hakudoushi/The Baby
Relationship: "brother"
Summary: Hakudoushi is another of Naraku's incarnations. He is later cut in half and reborn, one half remaining a baby which holds both Hakudoushi's and Naraku's heart. Hakudoushi treats Kagura much the way Naraku does, ordering her to take unnecessary risks with her life. While he is not necessarily always loyal to Naraku, he often uses his knowledge of Kagura's betrayal against her.
In the manga, Hakudoushi admits that, like Kagura, he also desires to betray Naraku. He and the Baby plan to use Moryoumaru to destroy Naraku; they wish to be thier own being and replace Naraku. Naraku eventually becomes aware of Hakudoushi's intentions and allows him to be destroyed by Miroku's Wind Tunnel/Kazaana.
Name: Moryoumaru
Relationship: "brother" (sort of...)
Summary:Moryoumaru is Naraku's most powerful minion. He is the result of Hakudoushi fusing different demons together over and over, until a seemingly perfect being is created. He was created as armor for Naraku's heart, and the Baby is hidden within him. Hakudoushi and the Baby plan to use him to destroy Naraku so they can become their own being and replace Naraku.
Eventually Moryoumaru is ordered to destroy Kohaku and take the shard from his back. Kagura saves Kohaku, but as a result of this Naraku becomes fully aware of Kagura's betrayal and no longer has any use for her. It is becasue of this that Kagura is killed.
Name: Kohaku
Relationship: companion/"friend"
Summary: Kohaku is Sango's younger brother. After being controlled by a spider-demon and killing his father and village, Naraku erased all of his memories. Naraku controls the boy by embedding a jewel shard in his back. While she does not like him at first, Kagura soon realizes they are in similar situations. Naraku is using them both, and they eventually become sort of friends as a result of this.
This relationship develops even further in the manga. Kohaku recovers some of his memory and begins to disobey Naraku, eventually confiding in Kagura that he is plotting to destroy him. Kagura eventually saves Kohaku's life, which ultimately results in her own death.
Name: Kouga
Relationship: enemy
Summary: Kagura makes fast enemies with Kouga, the leader of a wolf-demon tribe. By Naraku's orders, Kagura leads Kouga and his tribe to one of Naraku's castles so he can obtain the jewel shards in Kouga's legs. She traps them and then slaughters the majority of his tribe. As a result of this Kouga has vowed to destroy Kagura and Naraku to avenge the death of his comrades.
Name: InuYasha
Relationship: enemy (ally later on)
Summary: When InuYasha first meets Kagura, he believes her to be Naraku in disguise, as she has his scent. He later learns she is one of his incarnations, but that is more than enough to make an enemy out of him. Kagura, however, does not intend to harm InuYasha, as she hopes he can one day slay Naraku.
In the manga this relationship develops differently. After Kagura saves Kohaku, InuYasha and co. learn that she is trapped with Naraku, and has no choice but to obey him. They finally understand her, and even offer to let her join them, but she is unable to accept. When Kagura is dying, Inu-tachi rush to her aid, however they are too late.
Name: Sesshoumaru
Relationship: possible companion/love interest
Summary: Upon meeting Sesshoumaru, she declares that he has "a fine face". It becomes obvious that she develops a sort of infatuation for him. She believes Sesshoumaru to be unrivaled in power and strength, and that he is the only one with the ability to destroy Naraku. She attempts to get him to kill Naraku, but he refuses to kill him for her.
In the manga this relationship develops further. When Kagura is dying from Naraku's miasma, Sesshoumaru comes to her, and even attempts to save her with his Tenseiga. It does not work however, but she states that she is happy he was with her in the end. As the manga continues Sesshoumaru shows anger towards those responsible for Kagura's death. The usually calm Sesshoumaru even loses his composure and recklessly attacks Moryoumaru when he speaks badly of Kagura.