To respond to this excellent question you need to think about how the veil is interpreted by the parishioners of Mr. Hooper. It is clear from his encounter with Elizabeth that some see his black veil as a symbol of some form of secret sin that he has committed:
"Beloved and respected as yo are, there may be whispers, that you hide your face under the consciousness of secret sin. For the sake of your holy office, do away this scandal!"
And yet, at the same time, the horror that the black veil strikes in people indicates that when they are confronted with the black veil they are also, in a way, confronted with their own secret sin that they would rather forget about, ironically making him very successful in his job:
By the aid of his mysterious emblem - for there was no other apparent cause - he became a man of awful power, over souls that were in agony for sin.
Lastly, at his deathbed, we are told that Mr. Hooper himself regards the black veil as a symbol of the individual's isolation from God and from his fellow man because of his sinful state:
"When the friends hows his inmost heart to his friend; the lover to his best beloved; when man does not vainly shrink fro the eye of his Creator loathsomely treasuring up the secret of his sin; then deem me a monster, for the symbol beneath which I have lived, and die! I look around me, and, lo! on every visage a Black Veil!"
Thus Mr. Hooper interprets his black veil as a separation both from God and from others, but note how at the end he makes it clear that this is a condition that all humans suffer.
Therefore I think there are three main symbolic meanings in the tale. The symbol of the black veil is a very rich one because people interpret it differently, but it is important to ask - what do you as a reader think it symbolises?
The veil symbolizes either secret sin or the fact that people usually tend to hide things from others.