The characters are as follows:
The reverend is described as a gentlemanly person in his thirties, and a bachelor. He is introduced to us through the shock of his changed appearance; he has begun wearing a black veil. It is insinuated that the veil is a mark he chooses to bear as atonement for some great sin.
The Dead Woman
We know very little about the dead woman, other than that she was a young, and seems to have had some personal connection to Mr. Hooper. Her death clearly affects him, and the townspeople sense a sort of supernatural weight between them that lingers beyond her death, and coincides with Hooper's wearing of the veil. It appears that she, or something about her, is the reason for the veil.
Elizabeth is Hooper's fiance, and serves as a means for us to "get to know" Hooper. He may confide in her the answers to questions that the townspeople would find too forward or inappropriate. Even so, he does not acquiesce to her desire for him to remove the veil, and so she cancels their marriage, but remains his friend. Elizabeth serves as a way for us to see that Hooper is still human, and that his decision has been carefully considered; neither a deep personal love, nor the desire of his community, will affect his choice.
At the end of the story, Clark arrives at Hooper's deathbed in order to perform his last rites, speak of Hooper's virtuous life and excellent work as a clergyman, but also to ask that he reveal the secret of the veil before he dies; he is concerned that Hooper will carry a great sin with him to his judgment.
They serve as voices of reflection for Hooper's actions, such as by claiming that they had visions of the reverend and a girl's spirit walking side by side. Some of them are given names, such as Goodman Grey, but they do not personally factor into the story any more significantly. As a whole, they deepen the intrigue by allowing us to "see", rather than just be told, that Hooper's wearing of the veil is strange and unsettling.