The second movie, The Hobbit: The Desolation of Smaug, continues the story with Chapters 7 through 13 of the book.
A brief recap of Chapters 7 - 13 of The Hobbit
- Chapter 7, "Queer Lodgings": Gandalf leads the company to the home of Beorn, a skin-changer who can take the shape of a great strong man or a huge black bear. They rest a while, and Beorn supplies them with supplies and ponies to ride to the edge of the forest. Gandalf leaves them at Mirkwood; he had already told them that he had pressing business south and he had to leave.
- Chapter 8, "Flies and Spiders": The company enters Mirkwood and walk for days and days, growing weary and disheartened. Starving, they see (and smell) a feast being prepared by elves in the forest and attempt to join them, but the Elves vanish. The Dwarves are attacked and captured by spiders and, through cleverness and courage, freed by Bilbo. Then they discover Thorin is missing: when he stepped within the Elvish circle, Thorin fallen into a enchanted sleep and was later bound by the elves and taken to their king for questioning.
- Chapter 9, "Barrels out of Bond": The Dwarves are captured by wood elves and taken to a great hall in a huge cave where they are presented to the elven king. The Dwarves refuse to give the elven king any information and are locked up in cells. After a couple of weeks, Bilbo devises an escape for the Dwarves via the empty wine barrels the elves send back to Lake-town. Stuffing a dwarf in each one and tamping down the lid before the elves come to send off the barrels, Bilbo forgets that he has to account for himself and ends up riding a barrel, staying hidden with his ring. The barrels, collected by rafts-elves, are guided down the rivers to Long Lake and the town of men.
- Chapter 10, "A Warm Welcome": The barrels are pulled to shore and left there. Bilbo starts to get the Dwarves out of the barrels. They then find the Master of Lake-town, and Thorin declares his intention of reclaiming Erebor. The Master, while not believing Thorin for a moment, knows that it is politic to pretend to believe him, and Thorin's promise to share the riches is an inducement to feed, shelter and outfit the Dwarves (and Hobbit).
- Chapter 11, "On the Doorstep": Two days of rowing takes the company to the Lonely Mountain. After days of traveling on foot, they reach the Desolation of the Dragon. Reaching the mountain, they explore for days, trying to find the secret door. Finally they find it, but cannot get in. Durin's Day arrives and Bilbo, the only one who hasn't given up hope of getting in through the door, remembers what was said on the map that Elrond read. They are able to open the door...
- Chapter 12, "Inside Information": ...and Bilbo is the first one to go in. Balin goes with him but stops as the light from the door becomes just a faint outline. Bilbo goes down the tunnel by himself and encounters Smaug. The Dwarves, in the meantime, are on the ledge outside the door. Bilbo, in his conversation with Smaug, finally goes a bit too far with cheekiness, and Smaug sends a blaze of fire at Bilbo. Bilbo survives, although slightly singed, and returns to the Dwarves. He tells them of his talk with Smaug and then, uneasy, he persuades them to go inside the doorway into the tunnel. Barely have they shut the door when Smaug smites the side of the mountain, smashing wall and cliff with his huge tail. Smaug flies off to remind the Lake people who the real King Under the Mountain is.
- Chapter 13, "Not at Home": The company is trapped inside the mountain. Days later, there is nothing for it but to go down the tunnel to the great hall. Smaug is not there. Bilbo goes down by himself and finds the Arkenstone. The rest of the Dwarves, now that Bilbo survived and Smaug appears to be gone, come down and explore their way through the Front Gate and out of the mountain. Finding an old road, they end up camping in a rock-chamber, wondering where Smaug is.
A not-so-brief recap of the movie
Note: The text below is in two different colors. Black text (with a sans serif font, the page's default) indicates story material directly from the book. Blue text (with a serif font) indicates changes from the original book made by Jackson to provide greater character depth, background information, and entertainment.
Pouring rain inundates the town of Bree. Thorin enters the Prancing Pony and is joined by Gandalf. Gandalf tells Thorin he urged Thorin's father, Thrain, to unite the dwarves and take back Erebor. Gandalf is worried about the dragon and oh, by the way, someone wants Thorin dead. Thorin says he cannot unite the dwarves without the Arkenstone. Gandalf: "What if I were to help you reclaim it?"
Bilbo spies out the land and sees Azog and also Beorn in bear form. He rushes to tell the Dwarves and Gandalf what he has seen.
The company flees the orcs and wargs, running to a possible refuge; the Dwarves and Bilbo don't know is Beorn's house but think it is simply a place that Gandalf knows about. They manage to make it into the house and lock the monster chasing them outside. "What is that?" Gandalf: "That is our host."
Beorn's house is a marvel; all the furniture is sized to a man who must be immensely tall and large. Cattle live in the house, as well as goats, bees, and sheep. Horses whinny outside. Beorn patrols his land. The orcs are afraid of him and depart. Azog is summoned to Dol Guldur.
It's morning in Beorn's house. We see a huge chess set, some goats, and a huge bee. Bilbo wakes up; he is looking at the Ring. Faint murmuring Black Speech is heard.
In Dol Guldur, Azog talks with the Necromancer and sends Bolg after the Dwarves.
At Beorn's house, Beorn tells the company something about himself. His people were killed or enslaved (perhaps explaining the shackles on his left arm), and he claims that he is the last of his kind. He feeds them and lets them ride his horses to the edge of Mirkwood. The forest feels sick; Gandalf finds Black Speech graffitied on an old ruin. Galadriel telepathically tells Gandalf, "Beware the Necromancer. He is not what he seems." She sends him to the High Fells to find out what is going on. He leaves the Dwarves, who are surprised, dismayed, and upset with his leaving.
Upon entering the forest, the company becomes more and more disoriented, like the forest itself is exuding miasmic fumes. They lose the path then become ensnared by giant spiders. Bilbo, who had been climbing the tallest tree to see if there was a way out, is not captured by the spiders and sets about rescuing the Dwarves with the help of his Ring. His sword, in killing a spider, earns her name: Sting. Bilbo drops the Ring, though, and his actions show that corruption has already begun to set in.
Enter the Wood Elves. Led by Tauriel and Legolas, they get rid of the spiders and then capture the Dwarves. Bilbo, unseen because of the Ring, follows them to their home. The King of the Wood Elves, Thranduil, questions Thorin; Thorin, still angry about what happened 60 years ago in Erebor, refuses to give Thranduil any information and declines Thranduil's offer to help for a share of the riches of Erebor. He is taken to the dungeons and imprisoned with the rest of the Dwarves. Kili looks at Tauriel and falls for her — she had saved his life battling the spiders. Bilbo devises an escape for the Dwarves using empty wine barrels being sent to Lake-town. The escape works, to a degree, but the Wood Elves find out and follow them. Bolg and the orcs have tracked down the Dwarves and now attack. Kili is hit by a poisoned arrow. The Wood Elves choose to attack and kill orcs instead of capturing the Dwarves, thus the Dwarves escape. Thranduil closes his kingdom when an orc captive tells a massive army is preparing to attack. Tauriel, however, has left before Thranduil closed the gates, and Legolas decides to follow her.
Gandalf and Radagast go to High Fells; the tombs are empty. The Ring-bearing human kings of old are gone.
The company encounters Bard, a boatman of Esgaroth (Lake-town). Bard smuggles them into the town and outfits them with clothes and weapons. Bard is a descendant of the man who last ruled Dale, the one who defended Dale against Smaug to the best of his ability. Bard still has one black arrow — the only arrow capable of killing Smaug. Thorin talks with the Master and people of Lake-town, promising them a share of the wealth of Erebor if they help the company. The people of the town are delighted, and the company leaves, but not all go to the Mountain. Kili is forced by Thorin to stay behind; the arrow's poison is killing him. Fili refuses to go, choosing to stay with Kili; Oin, the company's healer, also stays with Kili. Bofur, sleeping off a night's drinking, doesn't wake in time to go with the company, so four dwarves are left in Lake-town.
Gandalf goes south to Dol Guldur. Radagast goes to warn Galadriel of the missing kings in the tombs. Gandalf fights with orcs but is ambushed by Azog. He escapes them but then the Necromander shows up — he is Sauron.
Thorin and his reduced company reach the Lonely Mountain. Bilbo discovers the hidden entrance after the dwarves have all given up. The key works, and they enter the narrow passageway of the mountainside. Bilbo is sent to find the Arkenstone and, while searching for it, wakes Smaug.
In Lake-town, Bard tries to take the black arrow to the gigantic bow that launches it. He is captured in the attempt, and his son hides the black arrow. Bolg and his orcs slip into town at night to find and kill the Dwarves. Legolas and Tauriel, though, also arrive and save the day. Kili is dying; Tauriel uses kingswort (pigweed) to save his life. They realize that loving each other will lead to a lot of trouble — this is all without words on Tauriel's part, though, while Kili is raving because he is so sick.
Legolas goes after Bolg, who didn't die with his orcs. In a cage, Gandalf watches the orc army, led by Azog, march from Dol Guldur.
The Dwarves go down into the mountain and find Bilbo. Thorin demands to know if Bilbo found the Arkenstone. Smaug attempts to destroy the Dwarves; the Dwarves rekindle the mountain's forge to destroy the dragon. Neither side is successful. Smaug leaves the mountain, with intention of destroying Lake-town and its people.
Changes from Book to Movie
I found a great little site where someone compared all the changes from book to movie: https://web.archive.org/web/20170608071819/http://gary.appenzeller.net/TheHobbitDiff.htm I like it because it is straightforward, not saying whether the changes are good or bad.
Changes from Book to Movie, as Noted by Me
Jackson and team, in Desolation of Smaug, effectively expanded the world of Middle-Earth to a depth and dimension that reflects the world of The Lord of the Rings books. Starting with a more somber tone, Gandalf meeting Thorin in Bree and informing Thorin of the price on his head, as well as giving Thorin information about his father and offering help, shifts the mood from that of the book The Hobbit — primarily a child's book, a "story" — to a saga of great adventure and danger.
The orcs are deliberately different from the goblins. The approach to Beorn's house, rather than handled with the book's dry humour, is dramatized with a tense chase scene, albeit one with some humour, as Bombur outraces them all to the house.
The scene from the book showing the starving Dwarves trying to find the Elven feast and being continually thwarted is not shown. Thorin, in the book, is captured by the elves while the rest of the company except Bilbo are caught by the spiders. Bilbo saves the Dwarves from the spiders. The Dwarves are captured by the elves the next day and taken to the Wood Elf home. Thranduil is quite different in the book, too. He is more gentle and wise, and not so harsh to the Dwarves, even if he does put them in prison cells.
Tauriel is entirely made up by Jackson, as is the pursuit of the orcs and of the Wood Elves when the Dwarves escape. The Hobbit does not include Legolas either.
Bard does not come into the story until after the Dwarves have arrived at Lake-town. All of the company leave Lake-town to go to the Lonely Mountain.
Bilbo is the only person of the company to see or talk with Smaug. The whole fight between the Dwarves and Smaug is a Jackson creation.