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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 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 - when a man succeeds, 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 Qhorin Halfhand 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. When the rangers want to execute her, Jon objects and wants to spare Ygritte, saying that she surrendered. However, when a ranger draws his dagger, 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 and later saved her life, 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 reveals this was not his intent and his capture of her/saving her life was not for sex.
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). Morever, 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.
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 - but when he checked the arrow in her, he sees it was not his arrow in her. Ygritte dies in Jon Snow's arms, with essentially the same dialogue as in the TV series. Jon is certain he can save her and insists she is not going to die. Ygritte smiles at this, remembers 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 he hopes he'll never learn this information.
Afterward, throughout the novels, Ygritte is ever-present in Jon’s thoughts.