How It All Goes Down
We start out in Venice, Italy, land of love and water. We meet two guys early on: Iago and Roderigo. Iago, who's been taking money from Roderigo in some sort of "arrangement," is upset at "the Moor," a.k.a. Othello, our tragic hero. Othello is a general in the Venetian army, and he just chose another man, Cassio, to be his lieutenant. This angers Iago, who wanted the position for himself.
Iago and Roderigo decide to get back at Othello by making a nighttime visit to Brabantio, the father of Desdemona (a.k.a. the woman Othello has recently eloped with). When Iago and Roderigo tattle on Othello for marrying Desdemona without her father's permission, Brabantio rushes to his daughter's room and discovers that she is missing. According to the angry father, this must mean that "the Moor" somehow "tricked" his daughter into whatever the two of them are doing together.
Cut to Othello in the next day or so, who's hanging out with Iago and talking about his new wife, Desdemona. Trouble is brewing since Brabantio is a senator and therefore pretty influential. It's clear that he'll try to split the pair up. But Othello isn't worried. Since he's legendary in the Venetian military, he believes his service record will get him through just fine. He adds that he really loves Desdemona, too.
The conversation is interrupted by Michael Cassio (the guy who got the lieutenant position over Iago), who says the Duke of Venice needs to see Othello right away, because there's some military action going down in Cyprus. Before everyone can peacefully exit, Brabantio shows up with Roderigo and various henchmen, ready to kill Othello or at least maim him severely for having the audacity to marry his daughter. Looks like everyone is off to see the Duke and settle the matter.
Once we get to the Duke, Othello speaks in his defense: he says Desdemona was an equal participant in their courting, and there was no trickery involved. They're now very much in love and married. Our woman in question, i.e. Desdemona, finally arrives and confirms the whole story. At this, the Duke tells Brabantio to stop whining and sends Othello to fight the battle in Cyprus. Desdemona states that she'll come along, as do Iago, his wife Emilia, Cassio, and Roderigo.
Iago and Roderigo have a little conversation during which Roderigo complains about being lovesick for Desdemona, and Iago says he'll get them together as soon as they bring down Othello. Once alone, Iago reveals a rumor that Othello was having sex with Iago's wife, Emilia. (The rumor is totally untrue and it's not even clear that Iago believes it.) To get revenge, he'll take out Cassio and Othello by convincing Othello that Cassio is having sex with Othello's wife, Desdemona.
So our cast of characters gets transported to Cyprus, where instead of battle there's just a big party (long story, read your play for the details). We note that Cassio is a ladies' man, especially around Emilia. While on watch together, Iago gets Cassio drunk and orchestrates a fight between him and Roderigo.
Othello intervenes and fires Cassio for being belligerently drunk instead of doing his job. Iago then convinces Cassio that he should ask Desdemona to tell Othello to give him back his job. Once alone, Iago schemes more about how he's going to convince Othello that Desdemona is having an affair with Cassio.
Cassio talks to Desdemona and she agrees to try to convince her husband to give Cassio his job back. As Othello is seen approaching, Cassio slinks off, not wanting to have an awkward moment with the guy that just fired him. Iago (entering with Othello) notes how suspicious it is that Cassio hurried off like that. Once the two men are alone, Iago plants (and massively fertilizes) the seed of suspicion. Cassio, he hints, is having an affair with Desdemona. He warns Othello to keep his eye out for anything suspicious, like Desdemona talking about Cassio all the time and pleading for his job back.
Othello is so upset he gets physically ill. Once Desdemona is back, she tries to bandage his head playfully with the "special handkerchief" Othello once gave her, a symbol of their undying love, an heirloom from his dead mother, and eventually the cause of a whole lot of trouble—which is why we later call it "the handkerchief of death."
To make a long story short, Emilia steals the handkerchief for her husband Iago, whom we learn has asked for it repeatedly in the past. Iago plants the handkerchief of death in Cassio's room. Othello enters, and Iago furthers Othello's suspicions with the aid of various outright lies. When Othello learns about the handkerchief, he decides that Desdemona is cheating on him, and because of that, she has to die.
The next scene brings us to Othello arguing with Desdemona while Emilia watches. He wants to know where the handkerchief is and Desdemona, oblivious, wants to talk about Cassio. Fighting ensues.
Shortly afterwards, we meet Bianca, a prostitute who's in love with Cassio. Cassio gives her the handkerchief he got from Iago, and swears it's not a love token from another woman. Some time later, Iago sets up a conversation between himself and Cassio, in which he gets Cassio to speak provocatively about Bianca. According to Iago's plan, somehow Othello, hiding and listening in, will think Cassio's speaking of Desdemona. So while Cassio is saying, "Yeah, I gave it to her good," Othello is thinking, "I'm going to kill that guy."
To make matters even worse, Bianca storms in and throws the special handkerchief in Cassio's face, having discovered that it indeed belonged to another woman. She storms out, with Cassio following behind her. Othello rages for a bit, and Iago advises that he strangle Desdemona. The next time the couple interacts, Othello hits her in the face (in front of a messenger from Venice telling him he has to go back home). Shortly after that, Othello yells at his wife, calling her a "whore," a "strumpet," and lots of other hurtful names. Filled with jealousy and indignation, he eventually resolves to kill his wife.
Back on the other manipulation front, Roderigo is getting tired of Iago taking all his money and not delivering the goods (i.e., Desdemona), as promised. Iago tells him to cool his jets, and also to kill Cassio when the opportunity arises, which, according to Iago, will happen that night between midnight and 1:00 AM.
Meanwhile, Desdemona and Emilia are talking together, and Desdemona begins to act strangely, foreshadowing her own death. She sings of it, too. Emilia, meanwhile, defends the act of cheating on one's spouse, especially if there's a good reason for it.
Iago and Roderigo hang out, waiting for Cassio. Roderigo tries to stab Cassio, fails, gets stabbed himself, and looks to be in trouble until Iago sneaks up and stabs Cassio in the leg. Two Venetian gentlemen run in at the sound of Cassio's screaming. Iago pretends he just stumbled in himself, declares Roderigo to be the assailant, and stabs Roderigo to death before the man can claim otherwise. Bianca runs in and screams a bit, and Iago tries to pin the mess on her. Emilia enters and Iago weaves her a lying tale. He instructs her to tell Othello and his wife about the news.
Othello, meanwhile, kills Desdemona, just as Emilia enters the room. In this moment of confusion, Emilia reports (incorrectly) to Othello that Cassio killed Roderigo. Othello is furious to find that Cassio is still alive, as that was definitely not the plan. Emilia finally puts two and two together and realizes her own husband is the cause of everyone's tragedy.
As people pour into the room, Emilia outs Iago for being a rat. Iago promptly stabs his wife, but not so promptly that the truth can't come out first. Othello demands to know why Iago ruined his entire life, but Iago refuses to give him (and us) a good reason. The Venetian gentlemen decide to take Othello back to Venice to face his punishment for killing his wife, and Cassio inherits Othello's post in Cyprus. Othello, overwhelmed by grief, decides to end his life rather than live without Desdemona.