Kitty Shcherbatsky starts out Anna Karenina like the kind of high society girls that we see in Gossip Girl: she's excited to be attending a ball, and she's head over heels in love. What could go wrong?
Everything. At that ball she can't wait for, she expects her Prince Charming, Vronsky, to propose to her, but all he does is stare at Anna Karenina. She finally gets a proposal, but it's from some weird, overly intellectual, socially awkward guy (Levin). She turns down Levin and has to go to a German spa to get over the shock and heartbreak of losing Vronsky.
Kitty has a revelation at the German spa. She realizes that she can't try to be something she's not. She's not some kind of high society seductress, and she's also not a charitable lady like Varenka. She has to be true to herself: honest, caring, and perfect for Levin.
In Part 2, Kitty discovers the same thing Levin takes a lot longer to learn: she can't lie to herself and she can't lie to society; all she can do is live according to her heart. And once Kitty learns this lesson, she spends the rest of the novel as a model to Levin. (For more on Kitty's role in Levin's religious searching, check out his "Character Analysis," in the section "Levin and Faith.")
In Part 1, Chapter 13, and in Part 5, Chapter 18, Tolstoy describes Kitty as preparing for battle. But she's not going off to crack skulls. In the first case, she is preparing to head to that ball where her heart will be broken. In the second, she's getting ready to nurse Nicholas, Levin's brother, in his final illness. These two scenes tell us something interesting about what Tolstoy feels the "battlefields" of a woman's life should really be: marriage, nursing, and birth. Basically, everything that goes with family life is in the female domain. According to Tolstoy, a woman's true vocation is to be the heart of the home.
Kitty is Tolstoy's ideal of what a woman should be, because she lives according to he true nature. Take a look at this passage in Part 5, when she's nursing Nicholas:
At the sight of the sick man, [Kitty] felt pity for him. And pity in her woman's soul produced none of the horror and squeamishness it did in her husband, but a need to act, to find out all the details of his condition and help with them. As she did not have the slightest doubt that she had to help him, so she had no doubt that it was possible, and she got down to work at once. Those same details, the mere thought of which horrified her husband, at once attracted her attention. (5.18.2)
There are some key words in that paragraph—"woman's soul," "horror and squeamishness" (remember, that's what Levin first felt when he saw Dmitri, as well), and "not [...] the slightest doubt." We see here that Kitty's womanly instincts tell her exactly what to do when faced with a sick person, in a way that Levin does not.
(Note: this part probably seems really dated. Keep in mind that this opinion is Tolstoy's entirely. He had his own specific preconceptions about what being a woman is all about. But—he wasn't a woman, and he was writing in a time when gender roles seemed much more defined than they do now. We can read and understand what he's getting at without necessarily agreeing with him.)
Just as Levin eventually becomes an ideal man over the course of the novel, Kitty is also an ideal woman (after her German spa experience). And what makes her ideal is that—although she gets jealous and she and Levin fight—she's aces at what really matters for women. She marries for love (instead of for social propriety, which is what drives Anna and Karenin together in the first place), she nurses her brother-in-law carefully, and she has a bouncing baby boy whom she loves.
In fact, she's so good that she makes Levin even better than he might have been otherwise. In other words, she's got goodness to spare.