Puritans were a movement within Protestantism that attempted to "purify" the church, stripping away the accretions of Roman Catholicism, especially as they had developed in the Middle Ages and returning to a more "pure" form of Biblical Christianity.
Puritanism was by nature anti-clerical. Rather than having a "priesthood", defined as a class that was uniquely ordained for a special role by apostolic succession and offered the Mass as a sacrifice in continuity with the Levitical priesthood of the Old Testament, Protestant denominations had ministers, people appointed to preach and undertake other administrative and liturgical positions within the church but who had far less ecclesiastical authority than the Roman Catholic priesthood. Rather than being appointed by papal or episcopal authority, ministers were chosen by their congregations and groups of lay elders could have ministers removed from their posts.
Many Puritans emigrated from England to the colonies because they opposed the Church of England, which was the legal state church, and wished for religious freedom. That being said, the communities they founded often were far more religiously oppressive than the England they left behind, although many increasingly had to deal with being part of states which were religiously diverse and cities that were increasingly secular. Although theologically Puritans asserted the "priesthood of all believers" and a "sola scriptura" doctrine, in practice, charismatic preachers and local churches could be quite powerful and influential in local politics, especially in towns that had been founded as Calvinist communities.
Cotton Mather himself was unusual in his efforts to reconcile science with religion, arguing for such unpopular measures as the smallpox vaccine. Although he did believe in witchcraft and was influential in the Salem witch trials, he appears to have been a moderating force, trying to establish at least some limits on what sort of evidence was admissible.
Puritan ministers occupied a central role in their society. While Puritanism stressed the concept of the "calling," which claimed that all work was divine, the occupation of minister was particularly important. Ministers made it their purpose to interpret scripture for the people. Most Puritan sermons began with a scriptural exegesis, which explained the meaning of a particular passage of the Bible. They would then move on to explain how that passage related to the lives of the congregation. So while each Puritan was responsible for his own salvation, they approached God as part of a community of believers in which the minister was the spiritual leader. As a prominent scholar of New England Puritanism has written: "Prayer was a weapon available to all people. But the clergy, with special qualifications, had special responsibilities."
In a society that made no distinction between church and state, ministers were important civic leaders as well, though in theory Puritan congregations were autonomous from each other. In seventeenth century New England, church was a place where local concerns, ranging from outbreaks of drunkenness, quarrels between local families, Indian wars, and of course, on occasion, witchcraft, were discussed. Because Puritans understood everything that happened in the material world as the work of God, ministers played a crucial role in framing community responses to events. They were, in short, opinion-shapers, moral arbiters, civic leaders, and very active members of Puritan communities, especially in New England.