"Sword, check. Hair flowing in the wind, check. Piece of wood, check. There's no way I can lose."
Their fight is unfortunately very short; Azog, mounted on his white warg, makes quick work of the Dwarven Prince. The dwarves look on in horror; Balin screams, and Dwalin almost falls in his hurried attempts to rescue Thorin. Only Bilbo is able to regain his footing. As Thorin lays defeated, about to be beheaded by one of Azog's minions, Bilbo draws his own sword, collects his courage, and charges. The hobbit knocks Azog's lieutenant to the ground, and after a short grapple, stabs the orc to death.
Bilbo then stumbles to where Thorin's body lays, as the rest of the orc pack surrounds him. He swings his sword a few times, in a vain attempt to intimidate them. Azog gives the command to kill the hobbit, but the rest of the dwarves suddenly charge in before the orcs can attack. After a short fight, Azog personally directs his attention toward Bilbo. He is about to kill the hobbit when the moth returns to Gandalf.
The Book: There is no such scene.
What difference does it make? This changes the events of the book a great deal. This scene is not in the book, and adds a violent battle that somewhat changes the tone of the scene. At this point, Peter Jackson has taken a step beyond "adaptation" and is now using Tolkein's book as a springboard to tell his own story. Furthermore, it is not in Bilbo Baggins' nature to rush into a battle like this. He is represented as braver and more skilled with a sword than he should be, according to the source material.
My Opinion: I have to admit to some bias, here. I'm what many refer to as a Bilbo Baggins fanboy. Any change that gives him more screen time and makes him look better is a good change in my eyes. As long as they keep it in character, I'm going to like it. That said... I loved this change.
I understand that this is a controversial viewpoint to have. This whole scene smacked of "action movie," and "obvious videogame tie-in." I can understand rolling your eyes as a 60 pound hobbit tackles a full-grown orc (in heavy armor!) to the ground. I've heard a lot of griping about Bilbo's character arc being complete far too soon, that he's not supposed to be killing things until the spiders of Mirkwood. I'm not convinced. His character growth isn't completely finished, and I'm fine with Bilbo seeing battle at the end of the first movie, instead of halfway through the second. It gives a nice climax that, honestly, the movie needed; a truly faithful adaptation would have ended things on a boring, ho-hum note.
Look closely during this scene. Bilbo is fucking terrified. And, with one rather unfortunate exception (when Bilbo joins the dwarves in combat after defending Thorin), Bilbo is terrible with his sword. He's figured out which is the dangerous end, sure, but otherwise just sort of swings it around clumsily. He doesn't know what he's doing, and is clearly in over his head.
He's just lucky he doesn't drop the damn thing.
Those who look down on this scene because Peter Jackson has turned Bilbo Baggins into some kind of action hero have missed the point entirely. Bilbo is scared, unskilled, and completely out of his element. But he's the first to jump to Thorin's defense, even if it means facing half a dozen mounted orcs by himself. Bilbo Baggins has a lot of heart, and this scene displays it better than the book ever did.