Sesshoumaru has a younger half-brother named Inuyasha, and they share a bitter rivalry throughout the series. He is less open about his feelings when it comes to his parents. Their father died long ago, and Sesshoumaru's mother was never mentioned in the series, so it would be safe to assume that she also passed away. We can also predict that she was an inu youkai, since Sesshoumaru is a pure-blood. Much of the information about their father comes from stories and legends we hear in the manga. In 2003, the third movie Tenka Hadou no Ken (Swords of an Honorable Ruler) shed more light about their father.
Inu-papa the father of Sesshoumaru and Inuyasha
With the powers of the three great swords Tessaiga, Tenseiga, and Souunga in his hands, this Inu no Taisho (Dog General) ruled over the western lands and was feared even amongst the youkai. He later fell in love with a human, Izayoi, who gave birth to his second son, Inuyasha. But Inu-papa had sustained fatal wounds from a battle with the great youkai, Ryukotsusei, and chose to die fighting for the safety of his family against the human Takemaru.
Inu-papa is the one person Sesshoumaru respects. He addresses him with the formal and respectful chichi-ue, and is determined to surpass his father's legacy one day. In their final conversation, he asked his father to entrust the Tessaiga and Souunga to him, as they were the strongest and most destructive of the three swords that Inu-papa wielded. However, he refused because he knew how destructive he was. While it was a huge blow to his ego that his father left him with the Tenseiga and Inuyasha with Tessaiga, Sesshoumaru is slowly coming to terms with his father's wishes. He was not one to disobey and go against him, after all.
He is haunted by his father's words: "Do you have someone to protect?" Through the inheritance of Tenseiga, Inu-papa probably wished to teach his violent son some compassion. The Tenseiga is a healing sword that could save a hundred lives if mastered, but it seemed like a joke for someone like Sesshoumaru to have. Nevertheless, he kept the sword with him anyway becauses it was from his father.
When Sesshoumaru finally felt compassion for someone else's sake, Toutousai reforged the Tenseiga to become a weapon in chapter 410. It gained a practically invincible attack - the meidou zangetsuha, which cuts open a path to darkness and directly sends the enemy's body into the next world. While this attack seems to defeat the sword's original purpose, this fighting Tenseiga pleased Sesshoumaru quite a lot. It looks like Inu-papa didn't leave him with such a useless sword after all.
Inuyasha the hanyou half-brother
Now he has been reawakened by Kagome, the reincarnation of Kikyou, who traveled to his time through a magical well connecting the two worlds. Along with Shippou, Miroku, and Sango, they embark on a journey to collect the pieces of the shattered shikon jewel and destroy Naraku. This time, he plans to use the shikon jewel to become a full youkai.
Although the brothers constantly make death threats and provoke the other, they are brothers after all, and underneath that bickering, I believe they really do love each other - though they would rather die than admit that. The protection placed on their swords keep them from seriously killing each other anyway.
Inuyasha is a lot more compassionate than Sesshoumaru, and though he likes to talk big, he cannot truly bring himself to kill his own brother. For example, when Inuyasha first figured out how create the kaze no kizu attack in chapter 129, Sesshoumaru wasn't prepared for it and was seriously injured. Although the Tenseiga ultimately saved Sesshoumaru's life in the end, Inuyasha could have done a lot more damage had he truly wanted to kill him. In another example, when Mouryoumaru used Inuyasha's stolen kongousouha attack to try to kill Sesshoumaru in chapter 407, Inuyasha hurries to free his brother from the death grip, because he didn't want the fact that Sesshoumaru was crushed by his stolen attack on his conscience.
As for Sesshoumaru, there were plenty of times when he came at Inuyasha with the true intention to kill him, its just that there is always an interruption. In chapter 187, after he visited the old tree Bokusenou and learned that Tessaiga also served to keep Inuyasha's youkai blood in check, Sesshoumaru went to verify Inuyasha's status himself. He not only held back in their fight, but he told Inuyasha's friends how to help Inuyasha return back to normal. He claimed he didn't kill him because there was no value in killing Inuyasha in that state, but Kagome speculated it was more like Sesshoumaru came to stop Inuyasha from going out of control. After that encounter, he seemed to have stopped pursuing the Tessaiga, as he realized Inuyasha needed it more than him.
In the third movie, they had to work together to destroy Souunga, however they were both being incredibly stubborn and were unsuccessful in their individual attempts. When Souunga unleashed its deadly attack on them, Sesshoumaru instinctly pushed Inuyasha out of the way, saying he was in the way, and took the full impact of the attack himself. In the end, they finally put aside their bickering for a minute to combine their attacks to win.
As the series progresses, it seems as though they do not hate each other as much as they used to. They both have their priority shifted on the same goal after all, which is to destroy Naraku. The longer they are in each other's presence, the more they resemble brothers. In chapter 323, Inuyasha fights against Naraku in his father's tomb. When Sesshoumaru shows up and sees the damages Inuyasha made, he admonishes him for damaging their father's grave by giving him a good punch in the face. Unlike all their other fights, this one was purely out of brotherly instinct on Sesshoumaru's part, because as the elder brother, it seemed almost natural to take responsibility for the younger brother. Inu-papa would be so proud~