Othello is so upset he gets physically ill. Once Desdemona is back, she tries to bandage his head playfully with the "special handkerchief" Othello once gave her, a symbol of their undying love, an heirloom from his dead mother, and eventually the cause of a whole lot of trouble—which is why we later call it "the handkerchief of death."
To make a long story short, Emilia steals the handkerchief for her husband Iago, whom we learn has asked for it repeatedly in the past. Iago plants the handkerchief of death in Cassio's room. Othello enters, and Iago furthers Othello's suspicions with the aid of various outright lies. When Othello learns about the handkerchief, he decides that Desdemona is cheating on him, and because of that, she has to die.
The next scene brings us to Othello arguing with Desdemona while Emilia watches. He wants to know where the handkerchief is and Desdemona, oblivious, wants to talk about Cassio. Fighting ensues.
Shortly afterwards, we meet Bianca, a prostitute who's in love with Cassio. Cassio gives her the handkerchief he got from Iago, and swears it's not a love token from another woman. Some time later, Iago sets up a conversation between himself and Cassio, in which he gets Cassio to speak provocatively about Bianca. According to Iago's plan, somehow Othello, hiding and listening in, will think Cassio's speaking of Desdemona. So while Cassio is saying, "Yeah, I gave it to her good," Othello is thinking, "I'm going to kill that guy.