When Smaug attacked the dwarves' kingdom in the Lonely Mountain to settle down there, why was the Arkenstone left? It seems to be this item which makes the owner king of all dwarves, so why did they not take it with them, wherever they would go?
The prologue of "The Hobbit: An Unexpected Journey" movie shows king Thror running to his throne to retrieve the Arkenstone as soon as Smaug arrives at the doors of Erebor.
He manages to get it, but while he tries to escape he runs into the dragon, swimming into the pile of gold of the treasure, and falls. He loses hold of the gem, which tumbles among all the gold items.
Thror still moves to catch it again, despite the menace of the dragon and the sheer impossibility to find it, a needle in a haystack. Thorin arrives and grabs him, forcing him to escape and run away.
So, apart from the role of the Arkenstone as a claim of kingship, Thror did all he could (and probably more than he should have done) not to leave it behind.
(the video is from the extended version of the movie, but I think it was present in the theatrical version too)
Possession of the Arkenstone had nothing to do with Thror's status as king of the Longbeards, the most senior line of the dwarven kings. Nonetheless, I am fairly confident that, had the King Under the Mountain been able to get away with the Arkenstone, he would have. The ultimate reason that Thror did not take the gem with him was he and his son Thrain only barely escaped from Smaug's attack. As Thorin (who only survived because he was outside of the Lonely Mountain at the time of the attack) tells it:
The few of us that were well outside sat and wept in hiding, and cursed Smaug; and there we were unexpectedly joined by my father and my grandfather with singed beards. They looked very grim but they said very little. When I asked how they had got away, they told me to hold my tongue, and said that one day in the proper time I should know.
Thror and Thrain only just made it out of Erebor, with Smaug's heat being enough to burn their beards. The most logical conclusion was that they did not have time to take anything with them once they realized the futility of their position; they just made a dash for the secret door in the lower halls.