The Clash of Wills
Byakuya is the first real antagonist Ichigo, Bleach's hero, must face as he tries to save Rukia from execution. Their conflict, however, does not represent a clash of good vs. evil, for the captain is not driven by malice, and in fact Ichigo is the one disturbing the "law". It is a battle of wills, of knowing when rules are meant to be broken, and the substitute shinigami teaches Byakuya a thing or two about "family."
Opening His Eyes
As I have reinforced again and again, Byakuya is at odds with his obligations and chooses the system over his desires; he has been programmed since birth to do this. Meanwhile, Ichigo is one of the most family-oriented people in the show, protecting his two younger sisters and also getting along with his father. Their principles are so perfectly opposite that I couldn't have chosen a better match up, myself.
After what polls say is one of the best bouts in the series, Ichigo succeeds in overpowering not only Byakuya's sword, but also his will. When the captain attributes his behavior to duty, Ichigo states that had he been in the same position, he would still have chosen to oppose the law. This has a transformational effect on Byakuya, and in one of his humblest moments, he says:
Kurosaki Ichigo, your ferocity has broken my sword. [...] The battle is yours.
Their relationship has less clear development after this, but even Byakuya knows that it was Ichigo and this battle that set his mind right about what's truly important. In accordance with that debt, he treats Ichigo with a respect not many have been able to earn, and also grants him a notable measure of trust.