The thing about Anna Karenina is there are (at least) two plot lines here. And each corresponds to a different classic plot. So we're going to distinguish between Anna's story, a Tragedy, and Levin's story, a Quest. Let's start with Anna's story:
This is the point in the classic tragedy when the hero feels unfulfilled, and looks to the future for gratification. In Anna's case, she's married to a rather boring man named Karenin. Nothing is exactly wrong, but she is dissatisfied in her marriage. Then she meets an attractive young man named Count Vronsky who's caught her eye.
This is the moment when the hero has decided what to do (usually something wrong) and seems to be "getting away with it." This is a stage is complicated in Anna's case: she does start out on an affair with Count Vronsky in Part 2, but she resists it for a long time, and feels guilty about it once it happens. In a way, part of Anna's particular tragedy is that she does something wrong, and doesn't truly enjoy having done it, for the most part.
Things start to go wrong, and the hero starts to freak out. Honestly, this stage starts from almost the instant that Anna sleeps with Vronsky. What troubles her (and Vronsky) is the necessity of deceiving Karenin. The longer she cheats on her husband, the more she despises him, even going so far as to shudder away from his touch when the two of them attend Vronsky's race together at the end of Part 2.
Just as it sounds, this is the point in the tragedy when everything seems to be turning against the hero. We'd pinpoint this stage's beginning at the moment when Karenin forgives Anna in Part 4, Chapter 17. After that, Anna and Vronsky travel in Europe, but Anna desperately misses her son, Seryozha. And when they return to Petersburg, she manages to sneak in to see him, but still can't reestablish her relationship with the boy. None of Petersburg's high society will be seen with Anna, and she grows more and more isolated and jealous of Vronsky's freedom. This affair is turning into a nightmare.
Tolstoy makes it easy for us to identify the "Death Wish" stage by giving Anna a literal death wish. With her total split from Seryozha, her loss of a social life, and her increasing suspicion of Vronsky, Anna becomes morbid and hateful. Her life feels so hopeless that she sees the entire world around her as sinful and oppressive. Finally, all that appears to be left to her is suicide: she begs for God's forgiveness and throws herself under a train.
Now let's turn to Levin's story: the Quest.
At this stage in the quest, the Hero feels that something is off in his life, so he has to set out to fix it. In Levin's case, the starting point of his quest is that he's 1) unmarried, and 2) frustrated with the lack of progress being made in his agricultural reforms on his estate.
Levin's journey is mostly an intellectual and emotional one (though he does go abroad for studies on the "worker problem" off the pages of the novel). He engages in numerous discussions with his companions, Koznyshev, Oblonsky, Sviyazhsky, and his neighboring landlords, about economy, social reform, and politics in contemporary Russia. On the romantic front, having been refused by Kitty, he pines for her from afar.
This is the point in the quest plot when the hero seems in sight of his goal, but there's still something in his way (like the Bridge of Death in Monty Python and the Holy Grail). For Levin, his quests to change Russian agriculture and to marry Kitty get intertwined. He expects that family life will give him all of the emotional and moral support he needs to solve the world's problems – but once he actually gets married, he realizes that it's not all it's cracked up to be. He and Kitty fight all the time, and Levin still hasn't worked out how to run his farm with the cooperation of the Russian Peasant.
Levin's at the end of his tether by Part 8: he loves Kitty and their son Dmitri, but he still can't overcome this sense of distance from both of them. He feels distant especially from his son, whose helplessness Levin resents. Also, he's had to give up his plans for the Russian Peasant. Now he's running his estates with some reforms at the local level, but his high-flying studies of zoology and the worker problem all seem to have turned to dust. His final ordeal must be to resolve his own existential angst and live as best as he can.
Levin meets an old peasant whose passing remark gives Levin all the answers he needs: he decides to live for the good that he feels in his soul. Basically, it's like this peasant flicks Levin's internal "Existential Angst" switch to "Off." Levin now realizes that the most important things in his life are Kitty and Dmitri, and that the best thing he can do in life is support them and live to do good for the people immediately around him. He reconciles himself with the fact that he won't be saving The Russian Peasant any time. This leaves us with that satisfying feeling that the end of a quest narrative should have – that Levin's life is going to stretch happily and satisfyingly into the future.