The handkerchief is significant in Othello, because it holds special meaning to Othello, Desdemona , and Iago. For Othello, the significance of the handkerchief is that it belonged to his mother and it was "bewitched" with magic. The handkerchief held special magic and power. If the handkerchief was lost then the "magic" in the relationship of the one who is in possession of it is gone. To Desdemona the handkerchief as a special meaning because it came from the one she loves, Othello.
The significance of the handkerchief for Iago is that it becomes a tool or a prop for him when it comes to manipulating Othello and accusing Desdemona and Cassio. Emilia doesn't seem to realize the importance of the handkerchief when Iago woos her to get it. Cassio thinks that Desdemona has given it to him and he gives it to Bianca a Courtesan. When really Iago plants it in Cassio's chamber to cause more mischief. In act 4.1 on page 2169 Bianca comes back and has the handkerchief and tells Cassio that it must be another woman's token and she will not copy it. Othello realizes it is his and he becomes enraged and wants to get revenge on Cassio for having and giving it to Bianca. Othello also becomes enraged when he speaks and questions Desdemona about the situation.
It is interesting how a piece of cloth can mean such significance and have such power over the characters and cause such destruction over a relationship. If this play was not titled "The Tragedy of Othello, The Moor of Venice" it should be titled " The Path and Power of the Handkerchief"