All of it came pouring out of Brienne then, like black blood from a wound: the betrayals and betrothals, Red Ronnet and his rose, Lord Renly dancing with her, the wager for her maidenhead, the bitter tears she shed the night her king wed Margaery Tyrell, the melee at Bitterbridge, the rainbow cloak the the had been so proud of, the shadow in the king's pavilion, Renly dying in her arms, Riverrun and Lady Catelyn, the voyage down the Trident, dueling Jaime in the woods, the Bloody Mummers, Jaime crying "Sapphires", Jaime in the tub at Harrenhal with steam rising from his body, the taste of Vargo Hoat's blood when she bit down on his ear, the bear pit, Jaime leaping down onto the sand, the long ride to King's Landing, Sansa Stark, the vow she'd sworn to Jaime, the vow she'd sworn to Lady Catelyn, Oathkeeper, Duskendale, Maidenpool, Nimble Dick and Crackclaw and the Whispers, the men she'd killed ...
"I have to find her," she finished. "There are others looking, all wanting to capture her, and sell her to the queen. I have to find her first. I promised Jaime. Oathkeeper, he named the sword. I have to try to save her ... or die in the attempt."  - Brienne VI (Chapter 31), A Feast for Crows, George R. R. Martin

By the middle of A Feast for Crows, it's incredibly clear that Brienne is completely in love with Jaime Lannister:

1. “The vow she’d sworn to Jaime, the vow she’d sworn to Lady Catelyn” - these are conflated in her mind now. She is finding Sansa Stark for Lady Catelyn AND for Jaime Lannister - and her vow to the living man, for whom Sansa Stark is “my last chance for honor,”  comes first. The quest for Jaime’s honor, because she knows him, because she watched him change before her eyes, and saw the man underneath his monstrous reputation, because he risked himself for her more than once: this vow is one that Brienne takes incredibly seriously. (Yes, of course, quite a few times during the course of Brienne’s inner monologue, she does feel compassion, and fear, and dedication to Sansa, but given that Brienne has never met Sansa, this is a more generalized compassion for others, which is certainly something Book!Brienne has in spades. Brienne is “beautiful” because she certainly can put herself in other people’s positions and she can imagine all of Sansa’s fear and anxiety in the face of the danger she’s in. I’m not saying that Sansa doesn’t matter to Brienne, but oaths matter too, a LOT, and the oaths are about Jaime and Catelyn.)

2. “I promised Jaime. Oathkeeper, he named the sword. I have to try to save her ... or die in the attempt.” When Brienne loves you, she will give her life for you. She dedicated herself to Renly’s service because he was nicer to her in public than any other non-related man had ever been; Jaime gives her a Valyrian sword and entrusts his honor into her keeping and vouchsafes HER honor to Loras Tyrell, which is like orders of magnitude greater than “being nice.” Over the course of their Riverlands journey, Brienne has learned that Jaime has a different side to him, that he was once just as concerned about honor, chivalry and oaths as she is; when he gives her Oathkeeper and her quest, he is entrusting her with something else as well: his fragile, newly rediscovered honor, and Brienne recognizes how precious this gift is (because she was in a sense the catalyst for his rediscovery of himself). In return, she will fulfill her quest or die trying because, well, Jaime and his honor mean that much to her.

3. “Jaime in the tub at Harrenhal with steam rising from his body” -  clearly SOMEONE made quite the impression on Brienne (it’s so funny in retrospect that Jaime thinks the “wench” is just being surly and gets in the tub with her in part to annoy her, and then we find out from Brienne’s POV that she was speechless because despite everything he’d been through, Jaime’s body was at least half godlike when he got into the tub with her. And she sure remembers that body.) I find it extremely amusing that she’s telling ALL of this to the Elder Brother, including her lustful thoughts about Jaime's naked body.