Briefly explain the importance of the scene with the handkerchief as it is depicted in Othello.

There is more than one scene in which the napkin is important. The handkerchief has become symbolic of Othello's commitment and love to Desdemona and plays an important part in Iago's scheme to convince Othello of his wife and Cassio's deceit. Othello specifically mentions its significance in Act 3, scene 4:

... That handkerchief
Did an Egyptian to my mother give;
She was a charmer, and could almost read
The thoughts of people: she told her, while
she kept it,
'Twould make her amiable and subdue my father
Entirely to her love, but if she lost it
Or made gift of it, my father's eye
Should hold her loathed and his spirits should hunt
After new fancies: she, dying, gave it me;
And bid me, when my fate would have me wive,
To give it her. 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.

Othello makes this statement after having asked Desdemona for the handkerchief to get rid of a 'salty and sorry rheum' which bothered him. When she extends a napkin to him, he asks for the one he gave her. He does this because Iago had worked him up to such an extent that he believed Desdemona and Cassio were having an affair and had told him that he had seen Cassio wipe his beard with the napkin. Furthermore, Iago had come in possession of the handkerchief when, on another occasion, Othello had been so worked up that he developed a headache and when Desdemona handed him her napkin, he pushed it away and it fell. Desdemona was more concerned about his condition than the napkin and left it.

Emilia picked up the napkin at the time:

I am glad I have found this napkin:
This was her first remembrance from the Moor:
My wayward husband hath a hundred times
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. 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.

Emilia gives Iago the napkin and does not realise what damning effect this act to please her husband would have afterwards. Iago plants the handkerchief in Cassio's room where he later discovers it. 

Iago has so influenced Othello through innuendo, suggestion and word-play, that the poor general is practically convinced that he has been made a cuckold by his wife and his erstwhile lieutenant. He had cleverly made the general suspicious by saying that 'I like not that' when the two of them came upon Cassio departing from Desdemona in a suspiciously furtive manner. When Othello enquired what he meant, he, at first, refused to say but later 'confessed' his suspicion at Othello's insistence. He told the general to be wary of 'the green-eyed monster' and that Desdemona had 'betrayed her father' suggesting that she might do the same to him.  

He later also conveys an occasion when he slept next to Cassio and the then lieutenant acted as if he were lying next to Desdemona, in the process kissing him passionately and expressing regret that she was with the Moor. Othello is not completely convinced and threatens Iago that if he is sullying her name without providing evidence, he will feel his terrible wrath. Iago deliberately asks the general if, as evidence, he wishes to see his wife perform a sexual act, knowing that Othello will be shocked. He then makes reference to having seen Cassio use the handkerchief and promises to provide Othello proof of the affair.

Iago later sets up a meeting with Cassio instructing the general to hide away so that he can surreptitiously eavesdrop. Iago manipulates the conversation with Cassio in such a way that Othello believes that Cassio is referring to Desdemona, when he is, in fact, talking about Bianca. The final, overpowering and most convincing event is when Bianca enters the scene and confronts Cassio about the handkerchief:

Let the devil and his dam haunt you! What did you
mean by that same handkerchief you gave me even now?
I was a fine fool to take it. I must take out the
work?--A likely piece of work, that you should find
it in your chamber, and not know who left it there!
This is some minx's token, and I must take out the
work? There; give it your hobby-horse: wheresoever
you had it, I'll take out no work on't.

Bianca is upset that Cassio had dared to ask her to some work on the napkin, which, she believes, is obviously the property of some or other flirt. This adds salt to Othello's already deep and open wound, that his precious gift should be treated as a piece of garbage. He sees this as undeniable evidence of Desdemona and Cassio's deceit.  

To make sure that Othello has gotten the message, Iago asks him later if he had seen the handkerchief and he assures him that it is indeed the object of Othello's affection. The general is absolutely overwhelmed and, encouraged by the malevolent Iago, suggests various ways in which he will murder his wife. He eventually decides to kill her in the very same bed in which she supposedly exercised her lust. The two men become co-conspirators and Iago promises to kill Cassio. Othello is pleased by his suggestion.

This series of events results in the tragic deaths of Desdemona, Emilia, Othello and Roderigo, with Cassio seriously wounded. The insidious master manipulator, Iago, is arrested to be dealt with later. Lodovico instructs:

... To you, lord governor,
Remains the censure of this hellish villain;
The time, the place, the torture: O, enforce it!