Conflict in literature involves struggles between opposing forces. These forces can be external, such as Man vs. Man; Man vs. Nature; Man vs. Society, etc. or they may be internal, such as Man vs. Himself (i.e. some weakness, fear, or decision with which the protagonist must wrestle).
Here are three incidents of conflict found in Ray Bradbury's Fahrenheit 451:
1. Mrs. Blake vs. Society
In Part One Mrs. Blake, the woman with a house full of books, is in conflict with a society that bans books. This society is represented by the firemen who come to burn her books. However, Mrs. Blake enters fully into this struggle against what she perceives as her repressive society by citing the words of a man who also refused to compromise himself. Like the Bishop of Worcester, whom she quotes, Mrs. Blake is willing to die for what she believes in and loves. As she strikes the match, she ends the conflict,
"We shall this day light such a candle, by God's grace, in England, as I trust shall never be put out."
2. Montag vs. Mildred
After witnessing Mrs. Blake's willingness to sacrifice herself rather than live without her books, Montag is powerfully moved. The next day, Montag wants to talk to his wife about what he witnessed at Mrs. Blake's. He asks her how it would be if he quit his job as a fireman.
"You want to give up everything? After all these years of working, because, one night, some woman and her books--"
"You should have seen her, Millie!"
"She's nothing to me; she shouldn't have had books. It was her responsibility, she should've thought of that. I hate her. She's got you going and next thing you know we'll be out, no house, no job nothing."
Montag senses the emptiness in his life after talking with Mildred, who accepts complacently the artificiality and restrictiveness of her world. Later, she even turns Montag in to the firemen. Montag loses in this conflict.
3. Montag vs. Society and the Mechanical Hound (Technology)
In Part Three, after Beatty realizes that Montag has kept books, Montag's house is burned and Beatty warns him that the Mechanical Hound will be "on the trail" of Montag if he tries to escape. After Beatty catches Montag with the buds in his ear, Montag turns the flamethrower upon Beatty, the Hound injects an anesthetic into Montag’s leg, but he manages to destroy the Hound with his flamethrower. He is then able to contact Faber, who tells him to follow the old railroad tracks out of town where there is a homeless people camp. Finally, Montag reaches the safety of the camp and others welcome him. When he hears on the news that a scapegoat has been used for the authorities to say that "Montag has been found," Montag knows that he is safe, and he has won in this conflict.