There are many minor conflicts in the story, but the main one, is absolutely a conflict of race.
I would say that the primary conflict is childhood innocence exposed to the evils of the world. This conflict also ties in with the title of the book and how mockingbirds represent innocence.
A secondary conflict could be racism; as the social hierarchy in Maycomb has the black community at the very bottom and they are often prejudged (i.e. Tom Robinson's trial).
One conflict of this story is the clash of White/blacks living in a small town and the ability to do what is right despite the influence of the town.