Secrets are the essence of this story.
The title takes its name from the veil that Hooper begins to wear without any apparent prompt; the thoughts of his parishioners immediately turn to the idea that it must represent some sort of secret that Hooper wishes to atone for by public display, but not to reveal. The nature of this secret provokes their curiosity, but as it becomes apparent that Hooper will not reveal what the secret is, or if there is any secret at all, the mystery begins to pass and Hooper is instead regarded with fear.
Finally, as he dies, Hooper suggests that everyone is wearing a veil. The exact meaning of this is a bit ambiguous, but it might be interpreted to mean that everyone is hiding some secret that they will admit to no one.
Without knowing what the secret is, or if there is one at all, the role that secrets play in the story is ultimately evidenced in our reaction to them; namely, curiosity, fear, bargaining (as attempted by several characters why try to get Hooper to explain or remove the veil) and distancing, as Hooper's veil begins to obstruct his ability to relate to others. Finally, Hooper himself has to point out that none of us are ever truly open with anyone, even God, and that our hidden secrets poison us against one another.