Bilbo’s increase in confidence and courage is the defining aspect of his character arc. At the beginning of the novel, he is shut away content in his Hobbit hole, happy to remain separate and coddled from the rest of the world. In the movie, Bilbo chooses to join the dwarves on their quest out of a sense for adventure. In the book, however, Bilbo wants nothing to do with them and only joins because he had been duped into signing a contract. Despite being on the adventure somewhat against his will, Bilbo still proves himself time and again, saving the dwarves from the spiders, the trolls, and a number of other dangers they encounter along the way. Bilbo’s character arc has even been cited as a sort of Christian metaphor for the growth of the soul, though how much Tolkien intended on this front is up for debate.

From a thematic standpoint, the ring of invisibility also plays into Bilbo’s brand of bravery. If Hobbits are the overlooked creatures of Middle Earth, then it is noteworthy that a ring which literally turns him invisible leads to so many of Bilbo’s most courageous moments. It is wrong to underestimate Bilbo. The world sees through him and looks over him—literally while he is wearing the ring—and yet, even people who look invisible to the outside world can display bravery in their own way.

Bilbo's development throughout the novel The Hobbit is one of the biggest factors in the plot. He begins as rather cowardly and much more of a follower than leader. However, throughout the story, he shows more courage and leadership as time goes on.

For instance, he shows great courage during his confrontation with Gollum in the cave—standing his ground and conversing with him, when the creature was deadly—and when he was at the bottom of a mountain full of deadly goblins. He showed leadership when he rescued his friends from the spiders in the forest later (using the very ring he found in Gollum's cave).

At the climax of the novel, he shows leadership and courage by being the one to enter Smaug's lair. He goes in and confronts the giant dragon, stealing evidence of the beast's horde of riches. This leads to Thorin retrieving his family's treasure and the dwarves eventually succeeding in their quest.

Arguably, Bilbo shows courage just by deciding to go on the quest in the first place. In Tolkien's mythos, hobbits are the everyman figures; they're homebodies who appreciate the good things in life (food, parties, etc.), and they don't really welcome change or uncertainty. For Bilbo, then, courage tends to be about pushing past his instinctive fears and reservations, and he does this throughout the novel—for instance, when he confronts the trolls in chapter 2, or (even more impressively) when he enters Smaug's lair in chapter 12. Bilbo is frightened enough in both instances to berate himself for ever agreeing to go on the quest, but his sense of duty and loyalty ultimately proves stronger than his fear.

With that said, the kind of courage and leadership Bilbo displays never quite map onto our conventional definitions of either. In literature—and especially in genres like fantasy or medieval romance—being a hero tends to mean being exceptional in some way (strength, power, wisdom, etc.). This, of course, is a better description of a character like Thorin than it is of Bilbo, and yet Tolkien portrays Thorin as deeply flawed (if still sympathetic). Thorin is brave and noble, and he generally observes a very strict behavioral code (think, for instance, of the formality of the contract he asks Bilbo to sign). Ultimately, however, he is much more susceptible to the corrupting influence of wealth than Bilbo. By contrast, Bilbo plays somewhat fast and loose with the rules; he's very good, for instance, at talking himself out of situations where a more traditional hero would fight (e.g., his interaction with the trolls, or the riddle game he plays with Gollum). This ability to think on his feet and bend the truth when necessary ultimately wins him the respect of the dwarves; by the time the group reaches the Lonely Mountain, Bilbo "has become the real leader in their adventure," with the dwarves looking to him for plans (chapter 12).

More even then cleverness, though, Tolkien suggests that it is Bilbo's humble and grounded sense of morality that makes him brave and a good leader. He steals the Arkenstone, for instance, not to keep it for himself but rather in an effort to prevent war (and save his newfound friends in the process). Thorin casts him out of the Lonely Mountain in response but ultimately acknowledges that Bilbo was in the right, suggesting that Bilbo's "courage" and "wisdom" spring in part from his valuing "food and cheer and song above hoarded gold" (chapter 18). Perhaps most significant, however, is Bilbo's decision to spare Gollum's life at a moment when it might have been safer or more prudent to kill him; although Gollum was blocking Bilbo's path out of the goblin tunnels, Bilbo's "pity" for Gollum prevented him from harming him (chapter 5). Tolkien's work was deeply influenced by his Christian worldview, so it is Bilbo's ability to show compassion, even at great risk or sacrifice to himself, that ultimately makes Bilbo a true hero.

What are some examples of Bilbo's leadership in The Hobbit?

During the first half of The Hobbit, Bilbo is just along for the ride.  It is not until Gandalf deliberately leaves the party to go through Mirkwood alone, that Bilbo is forced to take leadership. 

It begins when, after wandering from the path in the dark, the company become separated (Chapter Eight).  Bilbo, all alone in the dark, wakes up to find himself being wrapped up by a huge spider.  No one is at hand to rescue him.  He fights and kills it himself, with his sword Sting.  

Somehow the killing of the giant spider, all alone by himself in the dark without the help of the wizard or the dwarves [sic] or of anyone else, made a great difference to Mr. Baggins.

The next morning, Bilbo goes looking for the dwarfs and finds that they have been captured by a group of giant spiders and are all wrapped up, hanging upside down, waiting to be devoured.  He embarks on a long and harrowing rescue, using his sword and his magic ring, that ends up taking all day.  In the course of this rescue, he has to lead the dwarfs.  They are dazed and frightened, and also sleepy and weak from the spiders' poison.  Bilbo has to explain to them that "I am going to disappear" (for he had not yet told them about his ring), and get them to run away whenever he draws off the spiders.  They also have to make several stands whenever the spiders catch up to them, and they must turn around and fight.  This is the beginning of Bilbo's leadership. 

That evening, when they are finally safe from the spiders but have not found the path again, they realize that Thorin (the lead dwarf) is missing.  The next day, while wandering around trying to find the path, they are captured by a company of Wood-elves (Chapter Nine).   Bilbo adroitly slips on his ring, making himself invisible, and quietly follows the Wood-elves as they bring their prisoners back to their underground realm.  He sneaks in the gate, and thus begins his second episode of leadership.  Slowly, over many days, remaining invisible the entire time, Bilbo makes contact with all the dwarfs in their cells and discovers that Thorin is also being held in an isolated dungeon.  Then he conceives and carries out the famous escape by barrel.  To pull this off he has to convince all the dwarfs to come out of their cells when he sets them free, and to agree to get into the barrels.  They have to do this quietly, and with limited time before they are caught.  Bilbo succeeds in all of this. 

When the dwarfs arrive by barrel in Laketown, Bilbo gets them out of the barrels, listens to their complaints, and revives them.  Then, since Thorin is now with them, leadership immediately reverts to him, and Mr. Baggins once again fades into the background. 

The company make it to the Lonely Mountain and manage to find the secret entrance.  Now the dwarfs once again treat Bilbo like a hired hand.  Per their agreement, he goes alone into the tunnel, steals a golden cup, and brings it back to the dwarfs (Chapter Twelve). Then he tells them that although he has no idea how they are to get the huge mound of treasure out of the mountain, or in fact how they might get rid of the dragon, he is willing to make a second trip down the tunnel to see if he can find Smaug's "weak spot."

Naturally the dwarves [sic] accepted the offer eagerly.  Already they had come to respect little Bilbo.  Now he had become the real leader in their adventure.  He had begun to have ideas and plans of his own. 

After his second trip down the tunnel (during which Bilbo actually talks with Smaug), Bilbo returns to the dwarfs and they sit outside the entrance to the tunnel for a long time, talking.  Bilbo feels a deep foreboding and suggests they go inside the tunnel.   He has to urge them, but at last they agree to it.  Once inside the tunnel, he still feels fear.  He urges them to close the door.  They are reluctant, but he finally gets them to close it -- just before Smaug comes flying around the mountain and blasts with his fiery breath the mountainside where they had been sitting.  Bilbo has saved the dwarfs again.

After Smaug is killed (destroying Laketown in the process) and the dwarfs have taken over the Lonely Mountain, a standoff develops between Thorin, who has barricaded himself and the other dwarfs in the mountain, and the Lakemen, who are requesting that he give them a share of the treasure to make up for the loss of their homes.  Now it is time for Bilbo's final act of leadership.  He sneaks out of the Mountain by night and secretly offers a bargaining chip to the Lakemen.  It is the Arkenstone, the most valuable object in the hoard and the thing that Thorin most desires.  By doing this, he hopes to prevent a siege or battle, which is what seems to be developing.   

Bilbo's offer of the Arkenstone is not leadership in the typical sense, as no one is following him.  But he is taking independent, strategic action, putting into play an alternative solution other than the one offered by Thorin, the ostensible leader of the company.   And Gandalf (who now shows up again) seems to approve: "Well done!  Mr. Baggins!"