In Tolkien's Middle Earth, the Arkenstone is a rare and precious jewel found deep in the heart of the Lonely Mountain, centuries before Bilbo’s quest in The Hobbit takes place. It’s never explicitly explained within Tolkien’s books or the movie adaptations what the stone is made of. Readers know only that it is white in color, with glints of the rainbow glittering throughout its core.
The Arkenstone shone so brightly that it seemed to steal the light out of the room and turn it into something even more iridescent and radiant. It became the crowning jewel of the dwarven race, and was passed down from generation to generation of Durin’s folk. The stone can be seen displayed in pride of place above the throne of Thror, Thorin’s grandfather, during Peter Jackson’s film adaptation of the story.
The Arkenstone gave whoever held it the right to summon the seven armies of the dwarves from across the land, who had sworn an oath of fealty long ago. This is why it was not only considered a precious gem, but also a tactical weapon to protect the dwarven kingdom from invaders. This is how it became so closely linked with all the hoards of gold that Thror had begun to amass within the mountain. He saw the Arkenstone as a shield to keep them impenetrable and believed it allowed him to keep digging. The Arkenstone fueled the greed and ruthlessness that Thror was experiencing because he wore one of the seven dwarven Rings of Power, designed by Sauron in the Second Age to corrupt and twist this already stubborn race of beings. Spurred by the combination of the Arkenstone and the ring of power, it wasn’t long before Thror fell to dragon-sickness, and could no longer be reasoned with.
As a prelude to the narrative of the Hobbit, Smaug, the great and terrible dragon was drawn by the hoard of gold that had materialized in the Lonely Mountain. There, he burned alive all the dwarves who were living there, as well as the people in the nearby city of Dale. He then entered the mountain and, according to the films, settled into a long sleep among the coins, brooding over his treasure hoard until the already cursed gold was rife with dragon-sickness.
When Gandalf sends the dwarves to reclaim Erebor, he doesn’t realize quite how reckless Thorin will be when it comes to retrieving the Arkenstone. His desire for the Arkenstone is desperate enough for him to trade all the rest of the gold in the mountain for it. It's enouch that he's willing to risk Bilbo’s life for it, by sending him in to try to retrieve it from beneath the belly of the sleeping dragon. Even the idea of the Arkenstone, and being near enough to the mountain that he believes it is within his clutches, is enough to begin Thorin’s slow descent into the famed sickness of his forefathers. None of the other dwarves, nor Gandalf the Grey himself, could have predicted how quickly the Arkenstone and its greed would take a hold of Thorin.
There was one character who knew this though, right from the offset. Smaug seems aware of the effect that the stone had on Thorin’s grandfather, and almost relishes the idea of Thorin suffering because they cannot find it. In the movies, he cruelly taunts Bilbo:
“He sent you in here for the Arkenstone, didn’t he? Don’t bother denying it, I guessed his foul purpose some time ago. Well, it matters not. Oakenshield’s quest will fail. A darkness is coming. You have been used, thief in the shadows. […] The coward Oakenshield has wagered the value of your life and found it worth nothing.”
And unfortunately, within Thorin’s maddened and proudly entitled state, that’s not too far from the truth. In this regard, Smaug covets the Arkenstone because he knows that of all the wealth in Erebor, it is the one thing that the dwarves desire most. It is therefore the one thing that he wants to keep from them, because he enjoys watching them endure the agony of not having it. He enjoys seeing it tear them apart, and turn them against one another. He even says so himself:
“I am almost tempted to let you take it, if only to see Oakenshield suffer. Watch it destroy him. Watch it corrupt his heart and drive him mad… but I think not.”
Ultimately, it comes down to Smaug himself also having a form of dragon sickness. Like Thorin, and like Thror before him, he too is made greedy and unreasonable by the gold within the mountain. His famous line, “I will not part with a single coin, not one piece of it,” demonstrates that he would never freely give the Arkenstone up. Even though doing so would make the dwarven madness worse, it isn’t worth letting go of the treasure that he has hoarded over for all of those decades, not a single bit.