In The Scarlet Letter, what do the villagers fear and desire the most?

Although it's impossible to be sure, I think they fear that somehow Hester's behavior would become normative.  After all, isn't that one of  the basic purposes of all punishment, to set an example for others so that the behavior does not become acceptable?  It is interesting that, at the beginning of the book, it is the old biddies in the town who are loudest in their denunciations of Hester; the younger woman, perhaps because she is more likely to "sin" as Hester did, is much more sympathetic and forgiving.  And as the story progress, it seems that everyone is willing to forgive Hester as she becomes the "Angel" or "Able" one.

I suspect they most want the father of the child named for the same reason.  His punishment, perhaps alonside of that of Hester, would have sent a clear message to the townspeople that this behavior would not be tolerated in a "godly" town like Boston.  It is interesting that, excepting Chillingworth, of course, the people seem to lose interest in discovering the father's identity as they get on with their lives.  Maybe the will to punish is short lived ....

The villagers want Hester Prynne to be punished for her crime of adultery, but more than that, they want her to name her lover.  She refuses to identify the father of her child.  She is courageous in accepting full responsibility for the child.  

They believe that he must also be punished. For the Puritans, God was a harsh judge, so therefore, they were required to judge in God's name on earth.

"In religious terms it saw God not as a distant and harsh authority," 

"On the scaffold, Boston's highest clergyman, John Wilson, and Hester's own pastor, Rev Dimmesdale, each ask her to reveal the name of her partner in crime."

"Reverend Dimmesdale makes a particularly powerful address, urging her not to tempt the man to lead a life of sinful hypocrisy by leaving his identity unnamed. Hester refuses."

The Puritans feared sin.  They believed that a sinner within the community condemned them all.

"The Puritans required a strict moral regulation; anyone in the community who sinned threatened not only their soul, but the very possibility of civil and religious perfection in America and in England."

What do the villagers fear and desire in The Scarlet Letter?

The first two chapters of "The Scarlet Letter" have the longest passages describing the villagers of Boston. In the first chapter, Hawthorne paints a picture of a town that sincerely wants to be is a utopia, but it has found it must built both a prison and a cemetery. Thus, both sin, which the Puritans feared and death, which was inevitable, were active in the village. In the second chapter, the reason for the Puritans fear of sin is revealed during the first scaffold scene. Puritans believed that any sin committed in the community would cause God’s wrath to be visited on them. An illness or misfortune would show God’s disapproval. Puritans looked for sins that had been committed so that the sin could be brought out in the open and the members of the community could express their scorn for that sin. Consequently, Hester's sin is openly mocked and some women suggest her punishment is too lenient. As Dimmesdale alludes to when he questions Hester, the Puritans believed in predestination and public confession. This meant that the “saved” (those destined to spend eternity in Heaven) had already been determined. One knew whether one was a member of “God’s elect” by being able to avoid sin. Thus, Dimmesdale, as well as others in the town, would be mortified because his sin might mean he was damned, a Puritan's greatest fear, and not attain salvation, a Puritan's greatest wish.