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In the books
When a wildling man wants a wife (wildling or non-wildling), it is custom that he must capture her while risking a severe injury, or even death, by the woman and her relatives, and the woman must fight back. As a rule, the wildlings do not capture married women. As Ygritte explains to Jon - who is unaware of this custom - if the man succeeds kidnapping the woman, it shows that he is strong, brave and cunning, and this is what wildling women seek in men. If the man turns out to be an abusive husband, the wife can always slit his throat.
When Jon and the ranger Stonesnake capture Ygritte and her fellow wildlings during their ranging, Jon is unaware of this custom, and didn’t even know Ygritte was a woman at the time - believing her and her fellow wildlings to be men, until he sees her face after the capture. While they wait for the other rangers, Ygritte converses with Jon. She tells him the story about Bael, a former King-beyond-the-Wall, who kidnapped the daughter of Lord Stark. As she explains later, the story was meant to hint Jon what to do with her.
Some time later, Qhorin and the other rangers arrive. Ebben and Dalbridge want to execute Ygritte. Jon objects and wants to spare Ygritte, saying that she surrendered. Qhorin gives the task to Jon and simply tells him "to do what needs to be done." Jon can't bring himself to go through with it and he saves Ygritte's life, letting her go.
Because Jon captured Ygritte, Ygritte - who is attracted to Jon - explains the aforementioned wildling marriage custom to him and insists that they might be considered technically married because of it (according to loose wildling customs, which are not strictly defined "laws") and this stuns Jon. Jon explains this was not his intent and his capture of her was not for sex, but Ygritte is too stubborn. She tells Jon she expected him back then to have sex with her or to kill her, or both, and was confused that he did neither - not even after she told him the story about Bael.
At first, Jon rejects Ygritte's advances, not only due to his vows, but also because he does not wish to dishonor Ygritte and hurt her feelings, because sooner or later, he'd have to leave her and return to the Night's Watch (likewise, Sam felt bad about having sex with Gilly for the same reasons). Moreover, he may get her pregnant. He discusses this with Tormund, who fails to understand why Jon captured Ygritte if he does not want her. He explains to Jon that there is nothing dishonorable about free folk laying together and, in case Ygritte becomes pregnant - she can either drink moon tea, give birth to a strong son, or a lively laughing girl kissed by fire, and where's the harm in that? Jon has no answer. Later, when Mance grows suspicious of Jon's true motives, Ygritte assures him that they sleep together. This convinces Mance. Afterwards, Jon has no choice but to have sex with Ygritte. However, Jon and Ygritte fall in love and share a deep affection.
Jon feels ashamed, for the aforementioned reasons he had for initially rejecting Ygritte, and also for his feelings toward her. He is unsure what to do: to stay with Ygritte, he would need to become a wildling heart and soul; if he abandones her to return to his duty, the Magnar may cut her heart out, as a punishment for his treachery; if he takes her with him… assuming she will go, which is far from certain - no way he can bring her back to Castle Black to live among the brothers. A deserter and a wildling could expect no welcome anywhere in the Seven Kingdoms.
At one point, Jon tries to convince Ygritte that Mance Rayder cannot win: the wildlings are very brave, but they are undisciplined, and in battle - discipline beats valor every time. He fears that she and the rest of the wildlings will die. Ygritte is very angry to hear that, and implies that maybe he is still a crow in his heart. She tells him "You're mine. Mine, as I'm yours. And if we die, we die. All men must die, Jon Snow. But first we'll live".
After reaching Castle Black, Jon reports to Donal Noye and Aemon, openly revealing that he has broken his vows by sleeping with Ygritte - because Qhorin told him to do what it takes.
When Ygritte is mortally wounded, it is not clear whose arrow she was shot by, but it was not by Jon's. Unlike in the TV series, Ygritte was not shot in front of Jon while the wildlings and Night's Watch are trading arrow fire back and forth during the battle. In the novels, Jon finds Ygritte after she is injured and, for a frantic moment, he fears that it was his arrow which shot her; he checks the arrow, he sees it was not his (by the fletching), but feels as if it was. Ygritte dies in Jon Snow's arms, with essentially the same dialogue as in the TV series. Desperately, although it is obvious the injury is mortal, Jon tells her she is not going to die. Ygritte smiles at this, reminding him the cave they spent their first night in together, and Jon promises they will go back to that cave and again tells her she isn’t going to die. Ygritte cups Jon’s face, as Jon remains unable to accept her impending death, and she passes away in Jon’s arms. Jon is deeply grieved by her death. He does not know which of his sworn brothers killed her, and hopes he'll never learn this information.
When Alliser Thorne and Janos Slynt interrogate Jon roughly, they accuse him, among other charges, of oathbreaking by having sex with "the unwashed whore". Jon tells them he did what Qhorin commanded him to do, which was to do anything he could to gain the wildlings' trust, and that having sex with Ygritte was part of his ploy - but admits that he went beyond what he had to do, and cared for her, even though this angers his superiors further because it is a more serious breach of his vows. Jon knows well that many of the sworn brothers often visit the nearby brothel, but refuses to dishonor Ygritte's memory by comparing her to the whores of Mole's Town.
Afterward, throughout the novels, Ygritte is ever-present in Jon’s thoughts.