The first act of a story is the beginning, which stretches until the moment when the characters have gone to the point of no return. At the outset of the novel, Anna is bored with her husband, who takes no notice of her emotional life. So she takes up with someone who does have passion, Count Vronsky, while trying to keep her affair a secret for her husband and from Petersburg society. Levin, on the other hand, is frustrated by his lack of family and by his inability to get things done on his farm.
Act II starts when the characters seem as far from the resolution of their personal problems as they possibly can be. In Anna's case, this is the moment when she spills the beans to Karenin that she's having an affair with Vronsky. Soon afterwards, despite Karenin's efforts to the contrary, everybody in Petersburg knows about it. Once the secret is out, Anna and Vronsky begin to live together openly. Despite the fact that they don't have to lie anymore, Anna still feels guilty and unhappy.
Levin gets what he thinks is his heart's desire in Act II: he marries Kitty. But married life isn't as awesome as he expected, and he still has lingering doubts about why we're here on the planet. His existential angst hasn't been solved with marriage.
Act III is the point in the story when everything is resolved. Having been isolated by society and her own guilt and resentment, Anna throws herself under a train. Levin has a spiritual awakening that allows him to embrace his family life and the fact that he may not resolve the whole agricultural reform/Russian Peasant thing. He realizes what a great thing he's got going on with Kitty and his son, and resolves to "live for the good." By the end of Act III, the life that Anna has ruined has come to an end, while Levin has completed his quest for enlightenment.