The handkerchief is a gift that Othello gave to Desdemona. It has significant emotional value attached to it. It was a gesture of his love for her. When Emilia steals the handkerchief in Act III.iii, she comments on Desdemona's love for this object and the significance it contains for her:
I am glad I have found this napkin:
This was her first remembrance from the Moor:
My wayward husband hath a hundred times(325)
Woo'd me to steal it; but she so loves the token,
For he conjured her she should ever keep it,
That she reserves it evermore about her
To kiss and talk to.
This item is a symbol of Desdemona's love for Othello and Othello's love for her.
After Emilia steals this item, Iago plants it in Cassio's chamber to make it look like Cassio and Desdemona have been together. Iago reports:
I know not that; but such a handkerchief— I am sure it was your wife's—did I today See Cassio wipe his beard with.
This infuriates Othello that the item which served to begin their love may now indeed be that which ends it. This incidient fuels his belief that his wife was indeed cheating on him. This handkerchief also was from his mother and therefore had familial and cultural value to him.
The handkerchief serves the purposes of being a symbol of the love between Othello and Desdemona, as well as a pawn in Iago's plan to thwart Cassio's success.