A round character is a character that has complexity and depth, they undergo a transformation from the beginning of the novel to the end. Jim Hawkins is considered a round character because when he is first introduced to the audience he is just a boy scared of the pirate Pew but by the end of the novel he is a brave and confident young man who has saved himself from death as well as his fellow companions.
The audience can view Jim's dramatic change when he recaptures the Hispaniola tells Israel Hands to ask for God's forgiveness because "'[he has] broken [his] trust; [he has] lived in sin and lies and blood; there’s a man [Hands] killed lying at [his] feet this moment, and you ask me why! For God’s mercy, Mr. Hands, that’s why’" (239). Jim has grown so much on this voyage that he has the courage to scold a old sea dog as Hands for his crimes of piracy. In many ways Jim is more respectable and brave than the pirate crew. Further when Jim kills Hand's his emotional reaction is very mature as well, he doesn't like the fact that he had to kill another human being yet he swallows his action because otherwise he wouldn't be alive if he hadn't.
Flat Character
In contrast, flat characters are two-dimensional characters that don't have depth, however they are essential for the plots progression. Billy Bones is the perfect example as a flat character because his character and his involvement in the novel keeps him from being fully developed, however his death is the starting point for the building action of the novel that will lead to the climax. Without his death Jim wouldn't have gone to Squire Trelawney and Doctor Livesey who are responsible for the voyage to treasure island.