The handkerchief plays a major role in the play. It is probably the most crucial evidence to prove Iago's claim that Desdemona and Cassio, Othello's lieutenant, are having an affair.
To understand the handkerchief's importance in this regard, one also has to know its history as well as both its symbolic and sentimental value to our protagonist, Othello, general of the Venetian army. He tells Desdemona in Act 3, Scene 4 about its history, stating that an Egyptian charmer had given the napkin to his mother.
The charmer promised that the napkin would make her husband loyal and make her lovable. If she should lose it, her husband would turn against her and his eye would wander. His mother gave him the handkerchief on her deathbed and told him to give it to his wife if he should marry. Othello then said:
I did so: and take heed on't;
Make it a darling like your precious eye;
To lose't or give't away were such perdition
As nothing else could match.
Iago had created suspicion in Othello's mind that Desdemona and Cassio were having an affair by using innuendo and subtle suggestion. Cassio had earlier been dismissed by Othello for his involvement in a fracas which had been plotted by Iago and Roderigo. Iago had advised Cassio to ask Desdemona to petition Othello to reconsider his decision. This set the scene for Iago to develop his pernicious plot, for when he and Othello saw them together, he suggested that their liaison was unbecoming.
In Act 3, Scene 3, Desdemona starts pestering Othello with requests about Cassio's reinstatement. This irritated the general so much that he developed a headache. When Desdemona offered to wipe his brow with the napkin mentioned earlier, he pushed it away and it fell to the ground to be picked up later by Emilia, Iago's wife. She is glad about finding it and states:
I am glad I have found this napkin:...
...I'll have the work ta'en out,
And give't Iago: what he will do with it
Heaven knows, not I;
I nothing but to please his fantasy.
When she gives Iago the napkin, he immediately plots what he is going to do with it.
I will in Cassio's lodging lose this napkin,
And let him find it. Trifles light as air
Are to the jealous confirmations strong
As proofs of holy writ: this may do something.
The Moor already changes with my poison:...
In the same scene, Othello returns and threatens Iago that if he does not present ocular proof about Desdemona's illicit affair, he will kill him.
Villain, be sure thou prove my love a whore,
Be sure of it; give me the ocular proof
The sly and manipulative Iago then plays his trump card. He asks Othello about the napkin:
Tell me but this,
Have you not sometimes seen a handkerchief
Spotted with strawberries in your wife's hand?
When Othello confirms that it was his first gift to her, Iago says, in part:
...but such a handkerchief--
I am sure it was your wife's--did I to-day
See Cassio wipe his beard with.
He then says that if it should be the handkerchief, it would further confirm her guilt. Othello is enraged and promises to tear Desdemona to pieces. Iago allows him to rant and rave. Othello then kneels and takes a sacred vow of vengeance. Iago kneels with him and promises to offer his life in Othello's service. The malicious ancient has convinced the general that he is wholly on his side.
In Scene 4, Othello makes it a point to ask Desdemona for the napkin and she does not know where it is. She attempts to change the subject and asks him about Cassio's appeal, but this just angers the general even more and he insists that she procure the handkerchief. It is then that he relates its history. Othello superstitiously believes that the supposed charm has done its work. His outburst provokes Emilia into asking whether he is not jealous.
In Scene 1 of Act 4, Iago so plays on Othello's suspicion and jealousy that the general falls into a fit, which pleases the vindictive ancient tremendously. When Othello recovers, he tells him to hide for he is going to indulge Cassio in conversation, and he will have him tell about his affair with Desdemona. The sly Iago has already arranged to meet with Cassio and will talk about Bianca, a girl Cassio has been seeing. Othello, of course, does not know this.
Othello listens to the entire conversation, believing that every reference to Bianca by the two men is about his wife. The situation is worsened when Bianca later enters and waves the handkerchief at Cassio, exclaiming that she was insulted by him daring to ask her to 'take out the work' on the napkin, when it clearly was a gift from another woman.
Othello sees the handkerchief and Iago later confirms that it is, indeed, his. The general is overwhelmed by Desdemona and Cassio's perfidy and promises to chop her up into little pieces or poison her. Iago supports the general in everything he says and suggests that Othello not poison her. He recommends:
Do it not with poison, strangle her in her bed, even
the bed she hath contaminated.
Othello is convinced and has made up his mind. Desdemona must die. Iago promises him that he will kill Cassio. The scene has been set for tragedy to follow - Iago will have his revenge and innocents will die in the process.