Sesshoumaru is InuYasha'a older brother. He appears to be very cold and emotionless. Emotions are difficult for him, but his own journey (originally to hunt and kill InuYasha) will have very special plans for him.
He, too, possesses a sword given by his late father: this one forged from the right fang. It's known as Tenseiga, and there is an odd twist to it. Whereas InuYasha's Tetsusaiga (Tessaiga in Japan) can slay 100 demons in a single swing, Tenseiga's stroke can save 100 souls. It is a sword of life rather than death, and the cold Sesshomaru finds the sword to be quite worthless.
But Tenseiga sticks diligently to its destiny of saving lives. Once InuYasha managed to discover the secret to Tetsusaiga's power, a stroke from the sword almost killed Sesshoumaru, but Tenseiga stepped in and spirited Sesshoumaru away. The truth is that the unique swords are a twist of fate designed by his and InuYasha's father to ensure that neither brother can overcome the other.
For a while, Sesshoumaru couldn't make a weapon that could overcome Tetsusaiga, but then again, with Tenseiga in his possession, not even Tetsusaiga can strike him down. Tenseiga, however, will play a role in Sesshoumaru's personal journey, as he learns the truth about its powers of life.
In the meantime, Sesshoumaru finds another way to challenge InuYasha. Learning of the fallen demon Goshinki, who had managed to bite Tetsusaiga in two, he consults a twisted disciple of Toutousai named Kaijinbou, who takes a fang from the fallen demon and forges a new sword: Toukijin. Despite all the evil embodied in that sword, Sesshoumaru is able to overcome it and master the sword, finally gaining the chance to face his brother on equal terms.
In the manga, Sesshomaru receives a new sword, Bakusaiga, and his left arm is restored. It's explained in chapter 519 by Totosai that the sword is proof Sesshomaru can stand on his own as a daiyokai (greater demon) and has lost his obsession for his father's blade, Tetsusaiga. His arm returns for the same reason.
I'm told Sesshoumaru's name means "Destruction of Life" in Japanese.
Although the anime has not yet introduced Sesshoumaru's mom, she has appeared in the manga (though she has yet to be named).
I'm told a lot of fans have nicknamed Sesshoumaru "Fluffy" because of the fluffy thing on his shoulder. Many fans guessed that it was his tail or part of his cloak, and one fan contacted me saying Rumiko Takahashi said that it is actually a parasitic demon.
However, in the 2007 edition of "The InuYasha Profiles" the author, Rumiko Takahashi, clearly states the fur is part of his body. This can be further addressed by the fact that the mane also appears over the same shoulder in his dog shape. Sesshoumaru apparently has control over it, whatever it is, as in one episode he uses it to bind InuYasha.