Desdemona chats with the clown and asks him to bring a message to Cassio that he should come visit her. She wants the clown to make it clear that she's been good to her word about asking Othello for Cassio's reinstatement. She hopes all will be well.
The clown exits, and Desdemona is left with Emilia. She asks Emilia where on earth her handkerchief could've gone. Obviously upset to have lost it, Desdemona says that if Othello were the kind of guy to be jealous – which, of course, he isn't – her loss of the handkerchief would make him suspicious. She claims Othello has no such jealousy because the sun of his native land sapped it up from him.
As Othello approaches the two women, Desdemona declares that she won't leave Othello alone until he's agreed to see Cassio. Othello then enters the scene, clearly failing at his attempt to pretend that nothing's wrong.
Othello asks Desdemona for her hand, and notes that it is moist, which was thought to be a sure mark of a lascivious person. Othello waxes on about Desdemona's hand; he says it's a hot hand, which means it should turn to prayer and fasting and other chaste pursuits so it doesn't fall victim to the passions.
He then declares her hand is a frank one, and she interprets it to mean generous (for she says it's the hand that gave away her heart). Othello continues to riff on her perverted sexuality, as "frank" also meant "lusty" or "unable to conceal secrets."
Othello laments that in the old days, when one gave up their hand in marriage, they gave up their heart. This lengthy interlude is fun with wordplay, but also a scene setting to show Othello is suspicious, but not yet furious at Desdemona, who will walk into her own damnation.
To make matters worse, Desdemona changes the subject by reminding Othello that he promised to see Cassio (about getting his job back). This, of course, only inflames Othello's suspicions. He declares that his eyes are watering strangely, and asks if perhaps Desdemona has her handkerchief about her. As Desdemona hands him a normal handkerchief, Othello asks her where her special handkerchief is. Desdemona simply says she doesn't have it with her, likely because she doesn't want to upset him by saying that it's lost. So instead, she inadvertently upsets him by seeming to confirm that her handkerchief is with her secret lover.
Othello then tells her the story of the handkerchief. It's a family heirloom and totally sacred. A psychic informed his mom that, as long as she had the handkerchief, Othello's dad would love her. BUT, if she lost it or gave it away, Othello's dad would hate her and go back on the prowl.
Othello's mom gave the handkerchief to her son on her deathbed, telling him to give it to the woman he'd have for his wife. Othello explains this is why he told Desdemona to take care of the handkerchief, as losing it would be bad. Naturally, this worries Desdemona.
Othello asks point-blank if the handkerchief is lost. Desdemona hesitates, and then tells something of a lie. "It is not lost," she says (3.4.81). OK, so maybe an outright lie. The discussion over the handkerchief escalates into a huge fight. Othello keeps demanding to see it and Desdemona keeps refusing and telling him that he needs to forgive Cassio (which is really not helping her case about not being Cassio's secret lover). Eventually, Othello storms out, all for the want of a handkerchief.
Desdemona is shocked. She's never seen this side of her husband, and she doesn't know what's going on. Emilia declares this is no big deal, since women are like food to me.
Cassio and Iago choose this moment to show up, with Iago encouraging Cassio to make his case to Desdemona. She explains that now is not a great time for them to talk to Othello, since he's in bad spirits, and a little watery-eyed since he's lacking a handkerchief. Iago, all innocence, says that Othello's behavior sounds really strange, but being the good guy that he is, he'll try to find out what's wrong.
Desdemona slowly convinces herself that she shouldn't be angry at Othello for his behavior. Surely, she reasons, there must be some cause for it beyond the handkerchief, and probably something to do with Othello's work in Venice or Cyprus.
She decides that sometimes men, worried about big things, take it out on women, over little things.
Desdemona declares that she is to blame for getting upset at Othello over nothing. Classic enabling. Emilia, who, by the way, is totally at fault, says she hopes that Othello is just upset over things related to work, because it sure seems like he's jealous. "Heaven keep that monster from Othello's mind!" Desdemona prays (3.4.161), which means Shakespeare is repeating metaphors. Desdemona promises Cassio that she'll try again when Othello calms down, and she and Emilia exit, leaving Cassio alone in the scene.
Then Bianca, a prostitute of sorts and one who is in love with Cassio, comes in and yells at Cassio for not coming to see her often enough. Cassio makes weak excuses – it's clear that he doesn't take her seriously.
Cassio hands Bianca Desdemona's handkerchief, asking her to copy the pattern. Bianca gets all worried that Cassio has a new woman who's given him this as a token of affection, but Cassio explains that he found the handkerchief in his bedroom (thanks to Iago, unbeknownst to Cassio).
Cassio thinks surely someone will ask for the lovely handkerchief back, in which case he'd like to have his own copy, so could Bianca please take a break from being a prostitute and do some sewing for him, and also go away as quickly as possible?! After all, the last thing he needs is for Othello to see him with a prostitute.