In Othello, the handkerchief is quite significant. It is symbolic for faithfulness in marriage. It was given to Othello by his mother. She explained the meaning behind it. In Act 3, Scene 4, Othello relates that meaning to Desdemona:
Gave that handkerchief to my mother;
She cast charms and could almost read
The thoughts of people. She told her, that, while she kept
It, it would make her sweet and soften my father
Entirely to her love; but if she lost it
Or made a gift of it, my father's eye
Should see her as hateful, and his spirits should hunt
After new loves. My mother, dying, gave it to me,
And asked me, when my fate would have me marry,
To give it to my wife. I did so. And take heed about it.
Make it as darling as your precious eye.
To lose it or give it away is such complete destruction
That nothing else could match it.
No doubt, the handkerchief is important. It signifies faithfulness in marriage. That is why Othello is so upset. When he thinks that Desdemona has given the handkerchief to Cassio, Othello is enraged. He is consumed with jealousy. He cannot comprehend that Desdemona would do such a thing yet Iago has proof that the handkerchief was found in Cassio's home.
This is no ordinary handkerchief. It carries symbolic meaning. The message behind the handkerchief indicates that Desdemona has been careless and unfaithful in her marriage to Othello.
The handkerchief is important in 'Othello' because Iago uses it as part of a plan to encourage Othello's jealousy. Iago mentions that he heard Cassio (in sleep) mention that he (Cassio) once had an affair with Desdemona and that a handkerchief that Othello once gave Desdemona is not in the possession of Cassio. The trick works and Iago is now at the end of his feelings for his wife.