Varenka is a minor character in the novel, but she plays an important role. She presents an opportunity for Kitty's character development. Varenka is the adopted daughter of Madame Stahl. But, while Madame Stahl preaches religion but is actually quite cruel, Varenka is the real deal. She's good and self-sacrificing and charitable.
But while Kitty tries to imitate Varenka's goodness, Kitty realizes that she can't. When Kitty tries to get involved in the lives of a poor painter and his wife, she messes up everything. All Kitty can do is be herself. Varenka gives Kitty the opportunity to learn that the best thing she can do is to follow what her heart is telling her to do. For more on this, see Kitty's "Character Analysis."
Varenka, however, has her own tragic past. She was in love with a man and they were to be married, but his mother disapproved of Varenka, and so the wedding was called off. Her heartbreak over this ruined affair may be one reason why she seems so out of it all the time when she meets Kitty. She also has a brief flirtation with Koznyshev, when she comes to visit Kitty and Levin, but nothing comes of it.