Huck's conscience is a direct result of education, society, social norms, and what is considered appropriate in his culture. Most of the conflict revolves around the fact that he is skipping around the country with a runaway slave. Huck, his entire life, had been taught that slaves were not people but property, and Huck feels that way about Jim throughout a huge portion of the novel. In fact, he actually gets mad when Jim is excited about gaining his freedom and buying his family; Huck feels that Jim's family doesn't belong to him, but to the slave owner. So, this attitude is with him from the very beginning of the novel.
There are a few key events that start to turn Huck's attitude around. The first is when Jim declares that Huck is his best friend, just as Huck is about to turn him in. That makes him feel guilty. Another incident is when Jim is bitten by the rattlesnake; Huck feels bad about it and doesn't pull a prank like that again. Then, after Jim gets mad at him after the fog prank, Huck is truly sorry, and apologizes; this shows that he is starting to view Jim as a friend and equal, instead of property to be messed with.
Huck's true crisis of conscience comes when he discovers that Jim has been sold off by the duke and king. He tries to write a letter to Miss Watson, telling her where Jim is, but can't do it. He just can't bring himself to keep Jim in slavery, whether it is at the Phelps place or at the Watson place. So, he tears up the letter and in a telling statement says, "Fine, I'll go to hell then." Here we see him feeling evil for deciding to help a runaway slave, but not caring, because he considers Jim to be his friend. Huck risks his own safety and salvation to go get Jim. That resolves his doubts; he has a clean conscience, and moves forward from that point. By the end of the novel he is able to declare that he feels that Jim is "white on the inside," meaning, his equal.
I hope that those thoughts helped; good luck!