The ambiguity of the black veil is that its meaning is unclear to virtually everyone in Mr. Hooper's village. His first sermon wearing the veil gives some clues as to its significance: "The subject had reference to secret sin, and those sad mysteries which we hide from our nearest and dearest, and would fain conceal from our own consciousness, even forgetting that the Omniscient can detect them." Each member of his congregation begins to feel as though the minister had "discovered their hoarded iniquity of deed or thought." The veil, then, appears to signify the idea that each of us has secret sins, although we pretend, to all the world and even ourselves, to be sinless. Each of us, then, wears a figurative veil which we use to hide our true, sinful natures from everyone else; Mr. Hooper's veil is symbolic of this state. However, his congregation cannot fully realize this because, to do so, they would have to admit openly that they are secretly sinful, and this is the thing we most wish to hide. Although they all feel the same way, nobody knows it because nobody says it. They remain in doubt as to the precise meaning of the veil, even sending a delegation from the church to ask about it (although even those folks cannot bring themselves to just come right out and ask). Therefore, the veil's meaning remains in doubt until Mr. Hooper dies. His final words are the most explicit clue to the veil's meaning, but because the story ends with his words, we cannot know if his auditors understood.
The ambiguity of the black veil is that it covers the minister's face from both good and evil. Although he is mourning the lost innocence and kindness of the world overall, the reality is that he cannot be seen, and is not being heard. Therefore, any kind of assumption could come up as far as the minister's thoughts and feelings yet, they are impossible to be ascertained precisely due to the veil. The veil also represents detachment from reality, and separation from the rest of the world. The color is one and the same, it is solid, and it is meant to hide him. It is ambiguous precisely because of its purpose of hiding the minister and make him inaccessible to the rest of the congregation.