"Blue Eyes! Burst Stream!"
Yu-Gi-Oh The Movie

Seto Kaiba is truely cold person. He killed his step-father to take over Kaiba Corp. He stole Yugi's Grandfather's most treasured card, Blue Eyes White Dragon. He was willing to punish his own little brother for losing to Yugi.

Did I mention he's a multi-millionaire?

Kaiba has an undying passion for Duel Monsters, and because of this, he is Yugi's main rival. Yugi, well Yami no Yugi was the first Duelist who has been able to defeat him in a duel and since then, Yami Yugi has done it countless times.

Kaiba's goal is to humiliate Yugi for defeating him, he even had enough guts to kidnapp Yugi's Grandfather and force Yugi to enter his 'Death-T' Tornament to save him. He payed dearly for that...
Yugi crushed Kaiba's mind, making him comatose, after defeating him AGAIN at Duel Monsters. So Kaiba was out of the series for a bit until he completed his mind puzzle, in the Duelist Kingdom arc of the series. During that time his younger brother Mokuba took care of him.

When he was young, Seto challenged the President of Kaiba Corporation, Gozaburo Kaiba, to a game of chess when he came to the orphanage to adopt a child. As his prize for winning, Kaiba and Mokuba became his adopted sons. But, Gozaburo treated Kaiba cruelly to train his heir, so that's why Kaiba is so cold.

But Kaiba is not totally heartless, he loves his little brother dearly, he realized that when Mokuba cared for him in his comatose state. He despises Yugi with a passion, but will not allow anyone else to defeat him, he wants that honour for himself.