The handkerchief is first seen in Act 3 Scene 3, the scene containing the climax of the play where Iago begins to work directly on Othello, poisoning his mind.
The handkerchief is the strongest piece of 'evidence' of Desdemona's unfaithfulness that Iago uses. Iago tells Othello that he has seen Cassio using 'a handkerchief spotted with strawberries'. Since he has accurately described the handkerchief, Othello believes him and starts to doubt Desdemona. Hence, for Iago, the handkerchief is important because it speeds up the tragic events that follow, making his evil plans a success.
The handkerchief in the end, will lead to the death of Desdemona, Emilia and Othello. Desdemona will be killed by Othello who is convinced of her infidelity. Emilia will be killed by Iago for telling Othello and everyone else present that she was the found who gave the handkerchief to Iago after finding it on the ground. Othello will kill himself after realising that he has killed his beloved wife wrongly.
Desdemona's handkerchief is very special because it was given to her by Othello. It was HIS mother's handkerchief and it holds special significance to him, as well. Desdemona accidentally drops it and Emilia, Iago's wife, who is Desdemona's handmade, picks it up and flaunts it in front of Iago, who takes it from her. Iago plans to plant it in Cassio's quarters in order for Othello to find it, therefore providing "proof" that Cassio and Desdemona are having a torrid affair.