He is hoping that the desire for the Arkenstone would force Thorin to negotiate with the Men and Elves instead of fighting with them.
Bilbo believes that if the men and elves have the Arkenstone, which Thorin values even more than gold, they may be able to convince the dwarves to bargain with them. If an agreement can be reached, it will spare a good deal of bloodshed.
Also, Bilbo is getting quite tired of being trapped in the mountain, and wants to be somewhere with green grass underfoot and better meals.
Bilbo knows that Thorin will never give into the demands of the Men and Elves. He also see that it makes sense that the Dwarves compensate the men of the lake for the trouble the dragon caused, and probably felt a bit responsible since he was the reason the dragon stirred in the first place.
Bilbo himself wasn't sure at first why he kept the Arkenstone. Later he used it to eventually make peace with Bard and the other men.
He gave the jewel to Bard so that Bard could use it to force Thorin to give over a fair share of the treasure hidden in the mountain.
To make a truce with the Elvenking and the other man so that the fighting will stop.
He wants to kill him.
EveryDay Im Shuffling
He wants a truce between thorin and co. and the elves, because the goblins and the Wargs will be arriving to declare war.
Once Bilbo saw how greedy the Dwarves were, he knew the Thorin would do anything for the Arkenstone, even give up some of the treasure.
The Arkenstone of Thrain.
The dwarves are Bilbo's friends for most of the Hobbit.
He had possession of the Arkenstone.
The Arkenstone is the jewel of main focus in The Hobbit, and it's a prized treasure among the dwarves.
Bilbo hides the Arkenstone from the dwarves because he knew each one was greedy for the Arkenstone and they might kill each other or harm others for it.
The Arkenstone
Bilbo took the Arkenstone from Smaug's treasure trove. He gave it to the men to use as a negotiating tool.
Smaug has the Arkenstone in his hoard. Bilbo acquires it from the pile of treasure. He then gives it to Bard. Bard puts it with Thorin when he is buried. That is where it stayed.
Bilbo snuck out of the Dwarves camp and delivered the Arkenstone to Bard.
He forgives Bilbo for stealing the Arkenstone, and then tells Bilbo that everyone should be like the hobbits: more concerned with comfort than with treasure. Then Thorin dies. He is buried with the Arkenstone, and Dain, Thorin's cousin, is proclaimed the new king.
Bilbo took the Arkenstone. It was the treasure Thorin desired most.
When Thorin learned of Bilbo's treachery he kicked him out and said that the 1/14 share promised to Bilbo would be used to buy back the Arkenstone. He figured Bilbo would get little of it.
Bilbo takes the Arkenstone. He isn't entirely sure why he does it, but he picks it up and knew exactly what it was. He had some idea it might be of use later on.
Bilbo takes possession of the Arkenstone without telling Thorin.
Bilbo Baggins is a hobbit from The Shire.
The Arkenstone of Thrain.
Bilbo claimed that under the terms of the contract he was due a proportion of the dragon's treasure, and that he had taken the Arkenstone as his share.