After dinner, Othello suggests a walk with Lodovico and orders Desdemona to get ready for bed. He promises to meet her there soon, and demands that she send Emilia away. The men exit, leaving the women to chat and get ready for bed.
Emilia notes that Othello looked to be in better spirits, but she's shocked that he told Desdemona to get rid of her. Desdemona just shrugs it off – she can't risk upsetting Othello now. Emilia says she wishes Desdemona had never seen the man. But Desdemona responds that she loves Othello, so much that she would rather be with him, even when he's being totally strange, than live without him.
Desdemona is in a strange mood that foreshadows her coming death. She randomly tells Emilia that if she should die before her maid, she wishes to be buried wrapped in her wedding sheets. She then sings a song she learned from a maid of her mother's, who had been forsaken by her lover. She admits it was an old song, but it did well to bear out the maid's fate, as she died singing it.
Emilia tries to change the subject by noting how handsome Lodovico is, but Desdemona is stuck fast to this weird, mourning mood. She begins to sing the song about the willow, which is bad news, as willows are symbolic of disappointed love. (Remember in Hamlet, Shakespeare killed Ophelia off by having her "fall" out of a willow tree, mad with the double disappointed loves of her dead father and scornful Hamlet.)
So this song is essentially about a woman who excuses her awful lover because she loves him so much. The woman in the song doesn't blame him at all, but when she calls him a cheating jerk, he declares that the more women he gets with, the more likely she is to seek out other men. Desdemona sadly laments, "these men, these men!" (4.3.60).
She and Emilia then converse about whether women are ever as awful to their men as men are to their women. Emilia is certain this is the case, especially when it comes to cheating. Desdemona asks whether Emilia would ever cheat on Iago, and Emilia, much older and more cynical, tells her that plenty of women cheat. She says you could justify cheating in lots of different ways.
Desdemona declares she couldn't imagine ever doing such a thing, which leads Emilia to a bit of a rant. Emilia argues that women have the same need for sexual affairs. Since men change their women sportingly, women should have the same option.
She continues. Some men deserve to be cheated on; it's the husband's fault, not the wife's, if a woman has an affair. After all, she'd only be following the lead of her faithless husband. Desdemona bids Emilia farewell after listening to this sad speech from a sad woman whose husband obviously hates her and is now vindictively sleeping with other women. Desdemona hopes she can use others' bad behavior as a guide of what not to do, instead of an excuse to behave badly.