The handkerchief is the object that Iago uses as proof of Desdemona's betrayal: an embroidered handkerchief with strawberries, the first gift Othello gave her, which passes from hand to hand until Othello believes she gave it to Cassio, when she never did.
In Act III, Scene 4, Othello says that the handkerchief came from an Egyptian woman who gave it to his mother: as long as she kept it, she would always be loved; if she lost it or gave it to another, her husband would look at her with hatred. And he tells Desdemona: "There is much magic in those weavings." She, who had already lost it, answers that it was not lost.
The path of the handkerchief is this. In Act III, Scene 3, Desdemona drops it, Emilia finds it and gives it to Iago, who decides to throw it into Cassio's room and tells Othello that he saw Cassio wipe his beard with it. Cassio finds it and gives it to Bianca to copy, and in Act IV, Scene 1, Bianca throws it on the ground before Cassio, and Othello, hidden, recognizes it.
Alone, after picking up the handkerchief, Iago says: "These trifles", for the jealous man, "are signs as powerful" as Holy Scripture. In Act V, Scene 2, Othello cites the handkerchief to accuse Desdemona, and Emilia undoes the mistake: "I found it by chance and gave it to my husband." Cassio says that Iago confessed to having thrown the handkerchief into his room on purpose.