Sesshomaru is Inuyasha’s older half-brother, and he’s a full demon. The two share the same father, thus they look similar in appearance. He barely takes any interest in Inuyasha, because he despises humans, and is upset that Inuyasha is half-human and is soiling their father’s bloodline. He finally does take interest in Inuyasha, but only when he finds out that their father’s grave was hidden inside of Inuyasha’s right eye. Concealed in the grave is the Tetsusaiga, a powerful sword that can kill 100 youkai with one swing, left behind by their father and made out of one of his fangs. Inuyasha gets this sword while Sesshomaru gets the Tenseiga that has the ability to revive 100 humans with one swing. Sesshomaru doesn’t understand why he gets Tenseiga. He’s angered more when he finds out that only Inuyasha can hold the Tetsusaiga, because there is a barrier around it that demons cannot touch (without being injured/burned). Even though he hates humans with a passion, he does save the life of one human with his Tenseiga and that was a young girl named Rin. No one knows why he did this. Sesshomaru has some of the same attack styles/abilities as his half-brother Inuyasha, but his are more powerful due to the fact that he is a full demon. He can transform into his true form, a huge dog demon, which grants him more attacks and strength. Speaking of his huge dog form, When Inu-Yasha first got the Tetsusaiga, Sesshomaru was in his full dog form and Inu-Yasha used his new sword to slice off his left arm.
I believe that their father gave Inu-Yasha and Sesshomaru each sword for a reason. ‘Inutaisho’ gave Sesshomaru the Tenseiga in order to attempt to teach him to love humans. He gave Inuyasha the Tetsusaiga to not only keep the demon blood that runs through his veins in check, but to also make it virtually impossible for his two sons to kill eachother. In battle, if one of the two were to use their sword against the other, the other’s sword would protect them from the force of the blast. In doing so, making them unable to fight.