"Did you ever wonder why the men of the Night's Watch take no wives and father no children?" Maester Aemon asks him. "So they will not love. Love is the death of duty." Jon tells him that his father, Ned Stark, would do "whatever is right, no matter what," if faced with the choice between his family and his honor, and Aemon replies that most men are not Ned Stark: "What is honor compared to a woman's love? What is duty against the feel of a newborn son in your arms, or a brother's smile?"
The choice Jon makes in that moment -- the choice between his brother and his honor -- will haunt him for the rest of his life. After this, Jon spends a lot of time breaking the rules: he slept with Ygritte, fraternized with the wildlings, allowed them to cross the wall and ultimately join the Night's Watch, despite the protests of his sworn brothers. He later tries to amend things when he swears his fealty to Daenerys, refusing to break that vow even when Cersei offers them the support of her army against the White Walkers. For a while there it looked like Jon was the next Ned, his honor-blinders making it more and more difficult for him to see the big picture.
When it finally came down to it, the biggest decision Jon had to make was a decision he'd made a thousand times before. In killing Daenerys, Jon has to give up on love to do his duty to Westeros -- but he also has to ignore his duty to his queen to protect his family. Tyrion inverts Aemon's statement, saying that, when you think about it, "duty is the death of love." You can never have both.
The full quote, from George R.R. Martin's A Game of Thrones, ends with, "We are only human, and the gods have fashioned us for love. That is our great glory, and our great tragedy." Rhaegar Targaryen started this whole catastrophe by betraying his honor for love: marrying Lyanna Stark and fathering a child despite his duty to the realm. Jon Snow corrects this sin with a sin of his own, betraying his vows to save the ones he loves the most. The wheel of Westeros' great houses warring for power may have been broken at last, but the bigger wheel, the inevitable cycle of love and death and honor that Game of Thrones is built on, keeps on spinning into a faraway spring.