Days go by, and Bilbo and the dwarves don't hear a peep from Smaug.
Finally, they decide to walk down the passage to the center of the mountain to see what's going on.
But Smaug isn't home!
As Bilbo goes exploring over the treasure pile, he finds the Arkenstone, Thorin's most desired treasure.
The Arkenstone – a.k.a. the Heart of the Mountain – is a great gem that takes "all light that [falls] upon it and [changes] it into ten thousand sparks of white radiance shot with glints of rainbow" (13.19). Yes, please!
For some reason, he decides not to show it to the dwarves quite yet.
The dwarves arrive and start combing through the treasure.
Thorin gives Bilbo a present: a chain-mail coat made of mithril, Elvish metal.
Bilbo thinks, "I feel magnificent [...] but I expect I look rather absurd. How they would laugh on the Hill at home!" (13.37).
Bilbo reminds the greedy dwarves that they can't just hang around waiting for Smaug to get back.
Thorin agrees, and he leads them by memory through the old halls (and past some skulls and things) to the Front Gate.
Balin leads them up to an old watchmen's place so that they can keep an eye on Smaug (if he comes back).
So they go to sleep filled with wonder that there's still no sign of Smaug the dragon.