The Puritans were English Calvinists who believed that the Church of England had retained too many elements of Catholicism which they denominated "popery." They insisted that the church needed to be "purified" from these Cathollc elements, hence the name. They did not call themselves Puritan, the term was considered pejoritave; rather they called themselves the "godly." Although they eventually controlled Parliament in England and were responsible for the beheading of Charles I; during their early years they were frequently abused by Charles. They should not be confused with the "Pilgrims" who called themselves "separatists." The latter believed that the Anglican church was so tainted with Catholicism that it could not be purified and they must therefore separate themselves completely from it.
Those Puritans who migrated to America did so under the auspices of the Massachusetts Bay Company. A peculiar provision of the charter of the company was that there was no requirement that it keep offices in England. John Winthrop, a leader of the company, took advantage of this fact, and took the charter with him. This allowed the Puritans of New England to establish their own society without control from England.
Religiously, the Puritans were strict Calvinists who were firm believers in predestination. Those who had received the gift of God's grace (and thereby salvation" were known as "the elect." They tolerated no religious dissention, forcing Anne Hutchinson and Roger Williams from the colony, and hanging a group of Quakers who refused to leave.
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The Puritans were a group of people who grew discontent in the Church of England and worked towards religious, moral and societal reforms. The writings and ideas of John Calvin, a leader in the Reformation, gave rise to Protestantism and were pivotal to the Christian revolt. They contended that The Church of England had become a product of political struggles and man-made doctrines. The Puritans were one branch of dissenters who decided that the Church of England was beyond reform. Escaping persecution from church leadership and the King, they came to America.
The Puritans believed that the Bible was God's true law, and that it provided a plan for living. The established church of the day described access to God as monastic and possible only within the confines of "church authority". Puritans stripped away the traditional trappings and formalities of Christianity which had been slowly building throughout the previous 1500 years. Theirs was an attempt to "purify" the church and their own lives.