While Mr. Hooper doesn't explicitly state the reason he wears the veil, he certainly gives a number of clues.  He tells Elizabeth, 

"There is an hour to come . . . when all of us shall cast aside our veils . . . Know, then, this veil is a type and a symbol, and I am bound to wear it ever, both in light and darkness, in solitude and before the gaze of multitudes, and as with strangers, so with my familiar friends.  No mortal eye will see it withdrawn."

Mr. Hooper says that there will come a time when everyone will cast their veils aside.  The only time that we all definitively share is death, and so we must cast off our veils in death.  However, only Mr. Hooper wears an actual veil.  Therefore, we must assume that he suggests that we all wear figurative veils, while he claims that his is a symbol: both literal and figurative.  

Elizabeth responds, 

"Beloved and respected as you are, there may be whispers that you hide your face under the consciousness of secret sin."

In other words, there are rumors in the town that Mr. Hooper is wearing the veil out of recognition of some secret sin that he has and is hiding from everyone.  Mr. Hooper smiles at this, his "same sad smile" and says,

" . . . if I cover [my face] for secret sin, what mortal might not do the same?"

Thus, Mr. Hooper seems to acknowledge that his possession of secret sin could be the reason he wears the veil, and he suggests that every single person could likewise wear such a veil because all mortals possess such secret sinfulness.  Suddenly, the veil's meaning seems to dawn on Elizabeth, and she attempts to rush from the room.  Mr. Hooper cries,

"Do not desert me, though this veil must be between us here on earth.  Be mine, and hereafter there shall be no veil over my face, no darkness between our souls!  It is but a mortal veil—it is not for eternity!"

Mr. Hooper admits, then, that the veil will be removed when his soul has gone to God.  He only wears it while the mortal part of him lives, and he promises that it will not remain between them for eternity.  Therefore, we can piece together, as Elizabeth does, that the veil symbolizes the secret sin that each of us maintains, that acts as a veil between ourselves and everyone else while we live.  

The discussion between the Minister and his fiancee Elizabeth is the longest scene in the story and deals more directly with the question of why he is wearing his black veil than any other scene. He implores her not to abandon him, but he tells her that he is sworn to wear the veil for the rest of his life and cannot show his face even to her. He never gives her a candid explanation of what the veil symbolizes or why he is wearing it. He only suggests ambiguous possibilities which show by their phraseology that he either doesn't know why he is doing what he is doing, or else that he is not willing to reveal his secret even to her. For example, he begins two sentences with the word "If" and includes another "if" in one of those sentences:

"If it be a sign of mourning," replied Mr. Hooper, "I perhaps, like most other mortals, have sorrows dark enough to be typified by a black veil."

"If I hide my face for sorrow, there is cause enough . . . and if I cover it for secret sin, what mortal might not do the same?"

Obviously everybody in the story is going to be left guessing, and this suggests that Hawthorne fully intended to leave the reader guessing as well. The question most frequently asked about this story is "Why is the minister wearing that black veil?" But the real definitive reason or reasons will never be known.