What was going on between Bilbo and Thorin on that ledge?

From an interview with Richard Armitage (he plays Thorin):

• ... I think knowing that his father and his grandfather have been touched by this dragon sickness which doesn’t necessarily affect all dwarves, but some dwarves are susceptible to it.

  It’s this attraction to gold which becomes their downfall, has always been at the back of his mind.

  And I think the burden of taking his people back to their homeland, which is so massive, makes him a lonely figure, I think. Knowing that his grandfather failed, and his father failed, so if he doesn’t do it, there’s no other member of his line that will ever do this.

  [...] to play around with the dragon sickness [...] I think it’s going to be very interesting. I’ve looked at all sorts of different – I’ve looked at drug addiction, and along those lines, so that it actually has a physical effect on him, his mind and his body.

  But I think because he’s been a very heavy, melancholic character, I think the gold is going to change that, and it’s going to sort of bring him to life and make him the king that he should be, and more vibrant. But it comes at a price, I think.

Thorin's "dragon sickness" (For dragons covet gold with a dark and fierce desire) will become an issue in the third movie, at least according to Richard Armitage:

...we do get into those moments with the dragon sickness, later into movie three...

The Extended Version of The Hobbit: An Unexpected Journey contains a scene in which Elrond expresses doubts about Thorin to Gandalf.

• "Have you forgotten? A strain of madness runs deep in that family. His grandfather lost his mind. His father succumbed to the same sickness. Can you swear Thorin Oakenshield will not also fall? Gandalf, these decisions do not rest with us alone. It is not up to you or me to redraw the map of Middle-earth."

This is overheard by Bilbo and Thorin.

The altercation on the ledge is regarding the Arkenstone, which Bilbo was tasked with retrieving. When Thorin asks where it is, Bilbo does not answer him, which causes him to be enraged.

It's unclear why Bilbo didn't answer Thorin, choosing instead to try and leave the Hall beneath the Lonely Mountain. It could be because he felt the shame of failure, also that he felt escape from Smaug was a priority, but given the way he looks at Thorin, it's likely he detected something was amiss within the dwarf king.

Moments earlier, Balin reprimands the Thorin or not going after him, drawing attention to the 'Dragon Sickness' Oliver discusses above. When Balin warns him not to make the same mistakes as his grandfather, he replies:

Thorin: I am not my grandfather!

Balin: You are not yourself!

Thorin's judgement is obviously corrupted by something, be it his thirst for victory, the 'Dragon Sickness', his proximity to the lost gold or the Arkenstone itself.

Smaug earlier toys with Bilbo, telling him he is almost tempted to let Bilbo take the Arkenstone back to Thorin so he may watch him succumb to madness. We can infer from this the Arkenstone has similar properties to The One Ring. Thorin is acting not unlike Gollum when he is searching for his 'precious', enveloped in paranoia and anxiety. Perhaps Thorin believes Bilbo to have recognized the power of the Arkenstone and has concealed it about his person, trying to take it for himself. He could also have news of the Arkenstone that he is unwilling to share with Thorin.

Thorin's reaction is obviously inappropriate, but this is because his judgement is heavily impaired by a combination of the pressures upon him. A measured response is perhaps not to be expected given the circumstances.

And finally...

The incident seems to have been forgotten moments later.

This is because Smaug appears, rendering the intensity of the moment rather irrelevant. They simply need to flee...