The Arkenstone of Thrain.
The dwarves had begun to think Bilbo should put on his ring and scout the front door. He was getting tired of them and their expectations of him. And finally when the key-hole was revealed none of them thought to use the key! He had to shout at Thorin to get him over to the door in time.
well
The Elven-king, Thranduil, who ruled the Woodland Elves (Northern part of Mirkwood Forest) put Thorin and Company in prinson. Elves and Dwarves have distrusted each other ever since the First Age (The Hobbit is set toward the end of the Third Age), so Thranduil was not kindly disposed toward Thorin's Company. Thorin further angered Thranduil by refusing to tell him either his name or his business, because Thorin didn't trust Thranduil either.
Bilbo put a letter on the spoons to Angelica so they were for her
The Arkenstone of Thrain.
If you are referring to near the end of the book where Thorin rejects Bilbo, it is because of a dwarf's greediness for gold and treasure. Thorin had been searching for the Arkenstone of his father, Thrain, in the Lonely Mountain, and was upset that he could not find it. He was shocked when he found out that the very Hobbit that the dwarves had put all of their trust in had betrayed them and found the Arkenstone and did not tell him. Bottom line, dwarves are greedy. Hope this answer helped. :)
Bilbo had put on his ring, so he was invisible. Gollum couldn't find him.
The dwarves had begun to think Bilbo should put on his ring and scout the front door. He was getting tired of them and their expectations of him. And finally when the key-hole was revealed none of them thought to use the key! He had to shout at Thorin to get him over to the door in time.
He stole the guards keys, let all the dwarves out of their individual cells. Then he put them in barrels and let the elves throw them out of the "water gate" thinking the barrels were just empty and needed to be shipped to laketown.
well
That was the big mistake Bilbo made. There was no one to put him in it and close it up for him.
In presenting Bilbo as his hero, Tolkien was to a certain extent reacting to the overblown romanticised heroes of classic romance. Bilbo's values are very practical and down-to-earth. He (as all hobbits do) values peace, and comfort and good living. He therefore opposes Thorin's preference to make war on his neighbours rather than help them in their time of need. He even goes so far as to steal the Arkenstone to ensure that peace. His practicality is shown in his refusal of a share of the treasure so enormous that he could not possibly get it back to the Shire. But Bilbo is a hero all right. When the chips are down, and Gandalf has gone off to fight Sauron (called the Necromancer in the Hobbit), it is Bilbo who saves the dwarves' bacon. He shows courage, fighting skills, and ingenuity and is not deterred by extraordinary hardships and danger, yet he is not in love with the danger or with himself as a hero. All he really wanted was to get back to his hole in Hobbiton and put his feet up. Tolkien may have been reflecting on the kind of heroism shown by many of the soldiers in the First World War (his contemporaries), ordinary men who showed remarkable courage when put in a tight spot.
The Elven-king, Thranduil, who ruled the Woodland Elves (Northern part of Mirkwood Forest) put Thorin and Company in prinson. Elves and Dwarves have distrusted each other ever since the First Age (The Hobbit is set toward the end of the Third Age), so Thranduil was not kindly disposed toward Thorin's Company. Thorin further angered Thranduil by refusing to tell him either his name or his business, because Thorin didn't trust Thranduil either.
Bilbo put a letter on the spoons to Angelica so they were for her
The ring makes him invisible. The dagger is an elven blade and glows in the presence of orcs and goblins.
They escape in barrels sent downriver to be filled.