What is the Arkenstone worth to Thorin?To Bilbo? Why does Thorin want it?Why does Bilbo have and keep it?

The Arkenstone used to belong to his grandfather, it was found in the roots of the mountain, and it was faceted and polished by the dwarves. It is a part of Thorin's history.

Bilbo took it from Smaug's hoard of treasures, and although he's not quite sure why he keeps and hides it; he does use it later as a bargaining chip.