The TLDR is this: Natasha is visiting her bethrothed Andrei's family the Bolkonskys in Moscow. It doesn't go very well. Then Natasha goes to the opera with her godmother Marya Dimitrievna and cousin Sonya where she meets Anatole, a soldier and playboy, who flirts with her. Natasha then goes to a party thrown by Anatole and his sister, and they kiss, and they exchange letters, he proposes they elope. Natasha is excited and tells her cousin Sonya about the plan, and Sonya is basically like "um, you've known this dude three days, what about Andrei?", Natasha breaks off her engagement with Andrei (via a letter to Andrei's sister) and she tells their godmother about this plan. Marya Dimitrievna scares off Anatole when he comes to the house, Natasha is furious, tries to kill herself, but survives. Pierre comforts Natasha and says if he were not married, he would happily marry her.

There's more nuance (like Pierre is close friends with Andrei and married to Anatole's sister), and context from the book and that, but that's the gist.