I see it more often as a very emphatic NO! (: "Do you like this movie?" "Not at all! It's so boring."

It emphasizes 'NO', A: Do you like James? B: No A: You Don't? B: No, not at all.

What immediately came to mind for me was: - "Would you mind if I ..." (any reasonable proposal) - "No, not at all." By saying "not at all", the person indicates he/she really doesn't mind what's being proposed, putting the requester at ease. As Ellie said, a very emphatic no. Another good similar phrase: "absolutely not".

You could use 'not a bit' OR 'not (one) little bit' OR 'not in the slightest' instead of 'not at all'