What It Is
A handkerchief embroidered with strawberries, the first gift that Othello gave Desdemona. In scene 4 of Act III, Othello says that it came from an Egyptian woman who gave it to his mother with a promise: while she kept it, she would be loved, and if she lost it or gave it to another, her husband would look at her with hatred. The play makes the cloth a jewel worth whatever story is told about it.
The magical dimension of the handkerchief is Shakespeare's invention: in the source, Giraldi's novella, the object was only bait to set the farce in motion.
Where It Appears
In scene 3 of Act III, Desdemona tries to bind Othello's forehead, he refuses the handkerchief, and she lets it fall. Emilia finds it and hands it to Iago, who had asked her a hundred times to steal it, and Iago decides to throw it into the room of Cassio. Then Iago tells Othello that he saw Cassio wipe his beard with a strawberry handkerchief, and that this, added to the other proofs, accuses her.
Iago explains why an embroidered cloth is enough proof for someone who already wants to believe:
Trifles light as air / Are to the jealous confirmations strong / As proofs of holy writ.(III.3, ll. 330-331)
In scene 4 of Act III, Othello asks for the handkerchief, hears that Desdemona does not have it there, and reduces everything to one repeated word. Desdemona only wants to speak of Cassio, and Emilia notices that Othello seems jealous. At the end of the scene, Cassio gives the handkerchief to Bianca so she can copy the pattern.
Othello reduces Desdemona's request to a single repeated word, cutting off every attempt of hers to speak of Cassio:
The handkerchief!(III.4, l. 95)
In scene 1 of Act IV, Iago speaks of "handkerchief" and "lying" to Othello until he, repeating "the handkerchief, the confessions," falls into convulsions. Bianca throws the handkerchief to the ground before Cassio, irritated at having to copy it, and Othello, hidden, recognizes the cloth he gave his wife. Iago confirms that it was indeed Othello's handkerchief and, in the same scene, suggests strangling Desdemona.
Othello's speech breaks down into convulsion:
Is't possible? Confess? Handkerchief? O devil!(IV.1, l. 45)
In scene 2 of Act V, Othello tells Desdemona that he saw the handkerchief in Cassio's hand, and she answers that he found it, that she never gave it. Emilia undoes the deception when she says that she found it by chance and gave it to her husband. Cassio later says that he found it in his own room, where, according to him, Iago confessed to having thrown it on purpose.
It is Emilia who supplies the central piece of the plot:
I found by fortune and did give my husband;(V.2, l. 230)
What It Means
The handkerchief works as proof because jealousy is already prepared to receive it. In the play, the cloth does not convince Othello on its own: it merely cradles a conviction that Iago's insinuation had already manufactured without any object. The object changes hands several times, from Desdemona to Emilia, to Iago, to Cassio, to Bianca, and in each passage it takes on a new meaning without itself changing.
The handkerchief passes through many hands in four scenes and in none of them does it change: what changes is what each one says it is. The proof that Othello demands in Act III is made of the same material as the insinuation, and it comes undone only when Emilia says where it came from.