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In Harry Potter and the Prisoner of Azkaban, Sirius and Remus have the following conversation:

“Remus!” Pettigrew squeaked, turning to Lupin instead, writhing imploringly in front of him. “You don’t believe this ... wouldn’t Sirius have told you they’d changed the plan?”

“Not if he thought I was the spy, Peter,” said Lupin. “I assume that’s why you didn’t tell me, Sirius?” he said casually over Pettigrew’s head.

“Forgive me, Remus,” said Black.

“Not at all, Padfoot, old friend,” said Lupin, who was now rolling up his sleeves. “And will you, in turn, forgive me for believing you were the spy?”

“Of course,” said Black, and the ghost of a grin flitted across his gaunt face. He, too, began rolling up his sleeves. “Shall we kill him together?”

-Harry Potter and the Prisoner of Azkaban (Emphasis mine.)

Lupin's response of "not at all" seems to imply that Padfoot is not forgiven, yet the rest of the text implies that they both forgave each other and are now friends again. What does "not at all" mean in this context then?

Particularly in British English, a lot of responses are dictated by custom rather than literal meaning. For example, the "proper" response to "How do you do?" is "How do you do?" rather than an answer to the question.

In British English, "not at all" is used to mean that what the other person just said was not necessary to say. More commonly, it is a reply to "Thank You", meaning "what I did was not worthy of a thank you" (similar to the reply "it's nothing"). So to people accustomed to the interaction:

She: Thank you.

He: Not at all.

and it meaning that He is diminishing the importance of what She is thanking him for, it follows rather naturally that

Black: Forgive me.

Lupin: Not at all.

is Lupin saying that Black did nothing that would need Lupin's forgiveness. In other words, rather than forgiving Black for doing something wrong, Lupin is saying Black did not do anything wrong in the first place. There is "not at all" anything to forgive. This is not something that follows from the literal meaning of "not at all" but rather it comes from the way that specific phrase (idiom) is used in polite society.

As the OP points out, the literal meaning could be taken to indicate that Lupin does not forgive Black. I believe this is why he does not just say "Not at all" but rather "Not at all, Padfoot, old friend." By stressing familiarity and friendship, he indicates that he means the idiomatic interpretation, not the literal one.