In The Hobbit by J. R. R. Tolkien, Bilbo is a Hobbit on an adventure. At one point during the story, he must enter the lair of the dragon Smaug, underneath a mountain. When he enters, Smaug is sleeping. There is a huge pile of treasure in the dragon's lair. Bilbo, invisible because he is wearing the Ring, works up the courage to sneak out a single golden goblet. The dwarves outside are excited to see the treasure. However, when the dragon awakes, he can tell exactly what is missing. He flies outside and searches for the thief, sitting atop the mountain. Bilbo and the dwarves must hide inside the mountain. Smaug eventually returns inside and pretends to sleep. When Bilbo comes back in to attempt to find the dragon's weak spot, Smaug "awakes" and smells the intruder, who is once again invisible. Bilbo and the dragon engage in conversation, but Bilbo outwits the dragon with riddles and flattery, soon tricking Smaug into revealing his own weak spot. Specifically, he uses compliments to imply that he believes Smaug to be a great and indomitable dragon with not a single weakness. He convinces Smaug to show him the strength of the armor on his belly. Smaug is happy to show off, saying "My armour is like tenfold shields, my teeth are swords, my claws spears, the shock of my tail a thunderbolt, my wings a hurricane, and my breath death," and when he rolls over, Bilbo spies the telltale signs of the dragon's weak spot.

In The Hobbit, what is a "dragon-spell" and how does Bilbo find Smaug's weak spot?

The dragon, Smaug, proves to be a formidable threat to Bilbo and not just on a physical level. Although his stature is certainly frightening, he also possesses the power of "dragon-spell," which allows Smaug to control and command the behaviors and actions of those who meet his gaze. The impact of this ability on Bilbo is described in the book as follows:

Whenever Smaug's roving eye, seeking for him in the shadows, flashed across him, he trembled, and an unaccountable desire seized hold of him to rush out and reveal himself and tell all the truth to Smaug.

Bilbo must outwit Smaug, who has stolen the treasure of the mountain and exiled Thrór and his people, in order to help Thorin reclaim his land in the Lonely Mountain, his rightful treasure, and his throne. Bilbo remembers that his father once used to say that "every worm has a weak spot." Through strategic flattery, riddles, and a steadfast refusal to reveal his actual name, Bilbo weaves an air of mystery around himself. When Bilbo finally reveals that he has come not only for the treasure, but also for revenge, Smaug laughs him off. Bilbo announces that the underbelly of a dragon must be soft, and Smaug arrogantly shows him his underbelly to prove that this is not true, thereby revealing "a large patch in the hollow of his left breast as bare as a snail out of its shell."

With this information, Bard is later able to fire the Black Arrow into this uncovered spot in Smaug's breast, killing him and making way for Thorin and his men to claim their birthright.

The "dragon-spell" refers to the power Smaug has to hypnotize his enemies.  Smaug's eyes have powerful hypnotic qualities, and be looking directly at someone, he can convince them to do or say anything he wants.  Smaug, as a dragon, is in the same reptilian family as snakes.  Snakes have also long been described as having hypnotizing ability.

Bilbo cleverly tricks Smaug into announcing his weak spot by challenging Smaug's abilities.  He knows that the dragon is arrogant enough to want to prove how powerful he is; and also arrogant enough to assume that no one would be able to get at the weak spot.