Sesshomaru hates humans and cares little for other Youkai. This explains why he hates his brother so much. When Inuyasha was able to handle the Tetsusaiga his hatred only grew more. The Tetsusaiga having the power to kill 100 Youkai with one stroke. He was left with its compaion sword, Tenseiga. Tenseiga having the power to heal/revive 100 humans with one stroke. Since Sesshomaru despises humans he has no clue why the sword was left to him. Once he finds out that he can't hold or handle with Tetsusaiga and only Inuyasha can, the sibling rivary continues throughout the series.
As most of us can tell he shows very little emotion. As his father's successor, he's almost like a king of the western lands, which he governs.