Some years ago Tony Stark met Maya Hansen at a conference, and she told him about the Extremis process she was working on to cure severe injuries. (Prof Yinsen also met Tony there, as mentioned in the movie origin.)
A long-haired geek genius named Aldrich Killian tries to interest Stark in joining his think-tank Advanced Idea Mechanics. Tony brushes him off in favour of a one-night stand with Maya.
Now a smartened-up and rich Killian, head of A.I.M., comes to see Pepper Potts, who's now running Stark Industries as CEO. But again he gets a (more civil) brush-off.
Happy Hogan has been made Head of Security, and it's gone to his head.
Pepper goes home to where she and Tony are living together. Despite the fact that it's supposed to be their regular 'date night', Tony greets her as a remote-controlled Iron Man suit, because he's too busy creating the latest version of his armour (that comes to him when he calls it, in separate pieces that assemble on his body, rather than the walk-in suits he used previously). She's understandably not too happy about this.
(There's a thread running through this movie that scientist Stark was spooked by the mystical elements of the Avengers film, and by his near death then. And now he's retreated into his hi-tech basement.)
Meanwhile, there have been mysterious bombings with no sign of bomb residue, and a terrorist called the Mandarin has claimed responsibility by video.
Happy Hogan was suspicious of Killian but winds up following his bodyguard Eric Savin. He sees Savin meet another man who blows up. Happy is seriously injured so he's in no position to tell anyone that he saw a man explode.
After visiting Happy in the hospital, Stark challenges the Mandarin on national television, and foolishly gives out his home address.
Maya Hansen comes to see Tony at his residence to warn him about Killian and Extremis. But the Mandarin's men, led by Savin, destroy Stark's Malibu beach cliff home from helicopters.
As the mansion breaks up and slides into the ocean, Tony remotely clothes Pepper in his armour to save her, and she and Hansen escape. Stark then takes the not-completely-tested new armour back, but sinks beneath the waves.
Of course Tony survives and the JARVIS AI flies him in the suit to Tennessee where Stark has previously traced an early example of the explosions. The armour runs out of power, and Stark teams up with young geeky Harley to investigate the site. He tells the boy to call him the mechanic, although Harley obviously knows who he really is.
However, the Mandarin's minions don't want him to find out what's going on, and Stark is attacked by an Extremis-enhanced Ellen Brandt, who bursts into flame in front of him. Eric Savin is also there, and seems to be the only person who has undergone the Extremis process with long-term success. But Tony escapes with Harley's help.
Stark discovers that the Extremis process has been used on war veterans, who agree to work for the Mandarin if they survive.
Tony and the barely-functioning armour go to Harley's 'lab' in his family's barn. Stark works out where the Mandarin is operating from. He cobbles together some weapons from stuff Harley provides and buys at a Home Depot. Leaving the armour with Harley to recharge, Stark goes after the Mandarin.
Stark invades the Mandarin's place and manages to get to the very heart. There he finds that the 'Mandarin' is really an English actor, Trevor Slattery, who's being paid in money, beer and women. Tony gets captured and it turns out that Aldrich Killian is the real villain behind it all. He's taken Pepper and given her Extremis to persuade Stark to help him perfect the process (and save her). And Maya Hansen's in on it. But she has a belated attack of conscience, and gets killed by Killian for it.
Tony is left to think about it, chained up with some non-Extremis guards. But the Iron Man armour in Harley's barn sends a signal to let him know that it is powered up, and it's on its way. Stark expects it to arrive any minute, and starts taunting his guards. But the armour arrives in bits and pieces. Although it enables him to break free, he winds up fighting his guards with only one gauntlet and one jet boot.
Despite this, Stark is successful, and the last guard surrenders, saying he never liked working in this madhouse anyway.
Meanwhile, James Rhodes is still working for the US government, and his War Machine armour has been upgraded into the Iron Patriot armour. Killian left a false trail which had Rhodey sent to find Mandarin in Pakistan. But it was a trap, Extremis troops defeated him and took the armour.
Stark escapes with his complete armour, links up with Rhodes, and they go to prevent Killian's next plan which is to kidnap President Ellis. They contact the Vice-President about it, but he ignores the warning because Killian intends him to take over as President. The VP wants Extremis to cure his little daughter.
Savin dons the Iron Patriot armour and kidnaps the President from Air Force One. Iron Man chooses to save people who have been blown out of the plane rather than go after the villain, by forming a sky-diving daisy-chain.
Killian, still with the 'Mandarin' as a front, sets up the finale on a docked oil tanker that caused an infamous oil spill. He has the President suspended in the Iron Patriot suit, and is going to execute him on live TV.
Stark (armourless) and Rhodes infiltrate the site, but are soon being fired at and chased by Extremis-powered guards. Tony gets JARVIS to remote-control all of his armours and fly them into battle.
As part of the action, he'll sometimes get JARVIS to fly down a suit to temporarily enclose him.
Rhodey swings across like Tarzan and frees the President. Tony rescues Pepper, who has survived Extremis. But she falls to her death.
Stark tracks down Extremis-enhanced Killian, who exposes dragon tattoos on his torso, showing that *he* really is the Mandarin. Stark gets JARVIS to enclose Killian in one of the Iron Man armours, causing the villain to overheat and explode.
Killian survives this. But Extremis-enhanced Pepper also survived her fall. And it is she who finally does Killian in, with the help of parts of the latest Iron Man armour. Afterwards, she says she thinks she could get used to super-powers.
However, later Pepper gets de-powered, and Tony finally has his dodgy heart repaired. He throws the arc reactor that kept him alive into the sea at the site of his old mansion. And he gets JARVIS to destroy all the Iron Man armours in a firework display to prove his commitment to Pepper. And Harley gets some geeky presents from his friend the Mechanic.
Notes: The film started with a voiceover from Tony Stark. After the credits, we see that he has been telling the story to Dr. Bruce Banner, concentrating on his problems after the Avengers movie. Banner has been drowsing, in between reminding Stark he's not that kind of doctor (ie not a psychiatrist).