KAIBA 's ETERNAL RIVAL
THE BEGIN: MUTOU YUGI / PHARAOH ATEM
Yugi is Kaiba's eternal rival.
Their relation begins with the Blue Eyes White Dragon's steal. Kaiba steals it from Yugi because that card was the strongest of Duel Monsters and he absolutely wanted it, but Yugi retakes it winning a duel against him.
Since that day, Seto lives obsessed with Yugi and that defeat, for this reason his biggest ambition is winning against him in a Duel, even if it could be in many years. For Seto, that defeat is the prove that he isn't the n.1, and in order to become it again, he duels with his heart and soul day by day, trying to beat up Yugi and himself.
At the begin Seto seems to hate Yugi, but this isn't true, because Seto's only desire isn't Yugi's death, but Yugi's defeat. He wants to do it with his hands and his cards, and for no reason he will renounce to realize his ambition. In order to grant his wish, he organizes many duels and Card Battles that involve all duelists of Domino city, like the Battle City's Arc, where he wanted to appropriate of all the three God Cards.
Seto and Yugi's rivalry is the most important component of the series. Their duels are decided from destiny since 5000 years ago, where Kaiba was the High Priest Seto and Yami Yugi was the Pharaoh Atem. Kaiba's rancor and desire of revenge ride on that past life but he doesn't believe in the past, he wants to believe only in his present and future, because in his thoughts, life is the way that we decide to create and go through by ourselves.
Many times Atem tried to convince him that their duels depend by their past life and situation, but he never believed in his words. Kaiba wants to see the truth with his eyes and day by day, he starts to accept it and the meaning of the Millennium Items too.
Seto and Yugi are eternal rivals but many times they're partners in duels, like in Battle City, against Rare Hunters.
Seto doesn't believe in the heart of the cards but Yugi can persuade him to believe in his deck and not only in himself. Together they are the strongest duelist duo in the world (and Atem is very excited in duelling with Kaiba XD)!
Atem considers Seto his eternal rival like Seto considers him, but Yugi's idea of Seto's role in his life is a bit different.
Atem thinks about Seto principally like a duelist (and after it as friend), Yugi thinks about him as a friend and then as a duelist. Yugi desires to duel against him but in his heart their affiliation is not only rivalry but friendship above all. Seto is a VEEEEEERY proud and arrogant duelist and boy, but in the bottom (MUCH bottom XD) of his heart, he considers Yugi so important not only for their duels. Probably their past life marked them forever, in field and life.
Even anime's arcs like Alcatraz and Noah's World are prove of their important and special relation. By the way, Yugi and Kaiba's relation is the most important rivalry of the series and it highlights their opposite ideals, life-styles and ways to believe in themselves and their cards.