The main conflict in Fahrenheit 451 is Man versus Society. Montag is a typical citizen, living and working in a society that has bled individuality out of all personal interactions. People only connect with their television screens, not with each other, at least on any meaningful level. Montag, after meeting an individualist (Clarisse), begins to develop individuality himself, and finds himself stifled by the society in which he lives. Everyone around him is conditioned to act and react in a certain way, and Montag is pushing across the stream instead of going with it. He starts to truly realize this after he meets with Faber:
"You said get the money and I got it. I didn't really think of it myself. When do I start working things out on my own?"
"You've started already, by saying what you just said. You'll have to take me on faith."
"I took the others on faith! "
"Yes, and look where we're headed."
(Bradbury, Fahrenheit 451, Google Books)
Society, in the form of the firemen, Chief Beatty, and Montag's wife Mildred, all pressure him to give up his dreams of individualism and just go with the flow, live like everyone else, and forget that he could live life by his own standards and dreams. Instead, he pushes back so hard that he breaks his own life, escapes the city, and then moves on to try and influence society in a better, more positive manner.
The main conflict in Fahrenheit 451 is Man vs. Society, and this is presented through Montag's struggle against his oppressive, dystopian world. In the opening lines of the story, we see that Montag loves his job as a fireman and thinks there is nothing wrong with burning books. All of this changes, however, when he starts chatting with his neighbour, Clarisse, and he attends a fire in which a woman sacrifices her life. This sets the scene for the conflict because Montag realises that he is deeply unhappy with his life and starts to wonder if books contain some possible solutions.
The killing of Captain Beatty is a turning point in the story and one which brings this conflict to its climax. Forced to flee the city, Montag becomes a fugitive who bands together with the "book covers," a group of men who have memorised books for the benefit of future generations. When their city is destroyed by bombs, these men have the opportunity to rebuild a society in which censorship is no longer the status quo. As such, Fahrenheit 451 is a book in which Man conquers his oppressive society.