Sesshomaru is Inuyasha's older half-brother.  Unlike Inuyasha, Sesshomaru is full-fledged demon, and rotten to the core in my opinion.  Sesshomaru makes his first appearance on the show in "Aristocratic Assassin, Sesshomaru", where he is searching for the Tetsusaiga.  Since we all know who ends up with the Tetsusaiga, then there's no point in saying that he failed, miserably, at trying to get it.  Because Sesshomaru is all demon, he can't touch the Tetsusaiga, and thus, his goals to posess it are completely ridiculous and futile, for even if he did possess it, then he would not be able to use it anyway.  After ripping the "right black pearl" from Inuyasha's eye, Sesshomaru enters his father's tomb and there, he and Inuyasha fight, he coming off worse than his younger brother, who slices off his left arm just below the shoulder.  Yup, that's right folks, Fluffy's only got one arm.  Anyway, Sesshomaru later attacks Inuyasha again, still hoping to take the Tetsusaiga, but he can't because he sucks, and Inuyasha beats him again.  So, there you go Sesshomaru is a loser, and that's all there is, but he's still really strong.  As must as I hate to admit it, he does have some good points as well.  Firstly, he cares a great deal for an orphan human girl named Rin who tried to nurse him back to health when he was injured by Inuyasha in a fight.  He used his sword,

Tenseiga, the sword of life to restore her back to health. He also wears the most make-up out of any other character on the show, even the girls.  Later in the show, somewhere around episode forty-five or fifty, Sesshomaru also has a sword made.  THe sword is called Tokijin (I think) and it is evil.  It is made from the teeth of the monster who broke Inuyasha's Tetsusaiga.  At any rate, after his new sword is made, he no longer wishes for Tetsusaiga, and at one point actually saves Inuyasha with it.  Anyway, I'm far from Sesshomaru's biggest fan, but nonetheless, here you go, have some Fluffy.  Because I like to think of myself as being unbiased (hahahahaha oh god I'm going to die from laughing so hard), I should also point out that Sesshomaru does have some positive traits.  First of all he's taken under his protection, a little girl named Rin.  Rin was kind to him while he was injured after a battle with Inuyasha.  However, she was killed by Koga's wolf pack and when Sesshomaru came across her body, he brought her back to life.  There was no emotion displayed in this scene.  Afterwards, Rin just started following Sesshomaru and he takes care of her, usually relegating Jaken to babysitting duty.