There's a new guy in town
Prince Lev Myshkin comes to Petersburg. He has spent most of his life in a near-catatonic state and so has no idea of proper manners or society. Instead he is a purely good, completely honest man.
How do you solve a problem like Nastasya Philipovna?
As a former mistress, she is not really marriage material, but her seducer/tormentor wants to marry her off to the materialistic and ambitious Ganya, his employee. Also wanting to marry her are Rogozhin, who lusts after her, and Myshkin, who sees her as a doomed woman he could save somehow.
Too many suitors spoil the broth
Ganya can't quite pull the trigger and marry someone he despises. Nastasya ends up shuttling between Myshkin and Rogozhin, unsure of what to do. Rogozhin is torn apart by his jealousy. Meanwhile, Myshkin falls in love with Aglaya, and is torn between honor/duty and his feelings.
Myshkin and Aglaya are engaged. Enter Nastasya.
Myshkin and Aglaya's wedding would solve a lot of problems, since it is a precondition for Nastasya's marrying Rogozhin. But also it would create a ton of problems since Myshkin is socially wildly inappropriate for Aglaya, and because she is very jealous of Nastasya. In a climactic scene, the two women meet, and Myshkin seemingly rejects Aglaya in favor of his old promise to Nastasya.
When we return—Will Myshkin actually marry Nastasya?
Everything seems set for Myshkin and Nastasya's wedding, although pretty much everyone who has ever met Myshkin is dead set against it. Will Nastasya actually go through with the wedding?
Nastasya and Rogozhin—Till death doth them part
At the last moment, Nastasya ditches Myshkin and runs off with Rogozhin to his house. Where he kills her. Myshkin finds him with her corpse and they have a tense and half-crazy night together.
Unable to handle the events at Rogozhin's, Myshkin suffers a complete mental collapse, and goes back to his almost catatonic state. There is no hope of recovery.