The Pilgrims, also known as the Pilgrim Fathers, were the English settlers who came to North America on the Mayflower and established the Plymouth Colony in what is today Plymouth, Massachusetts, named after the final departure port of Plymouth, Devon. Their leadership came from the religious congregations of Brownists*, or Separatist Puritans, who had fled religious persecution in England for the tolerance of 17th-century Holland in the Netherlands.

They held many of the same Puritan Calvinistreligious beliefs but, unlike most other Puritans, they maintained that their congregations should separate from the English state church, which led to them being labeled Separatists. After several years living in exile in Holland, they eventually determined to establish a new settlement in the New World and arranged with investors to fund them. They established Plymouth Colony in 1620, where they erected Congregationalistchurches.The Pilgrims' story became a central theme in the history and culture of the United States.

* Brownist beliefs—true churches were voluntary democratic congregations, not whole Christian nations—as taught by Robert Browne, John Greenwood, and Henry Barrow. 

@ " Pilgrims", Wikipedia. https://en.m.wikipedia.org/wiki/Pilgrims_(Plymouth_Colony)

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Puritanism, a religious reform movement in the late 16th and 17th centuries that sought to “purify” the Church of England of remnants of the Roman Catholic “popery” that the Puritans claimed had been retained after the religious settlement reached early in the reign of Queen Elizabeth I. Puritans became noted in the 17th century for a spirit of moral and religious earnestness that informed their whole way of life, and they sought through church reform to make their lifestyle the pattern for the whole nation. Their efforts to transform the nation contributed both to civil war in England and to the founding of colonies in America as working models of the Puritan way of life.

Theology

Puritanism may be defined primarily by the intensity of the religious experience that it fostered. Puritans believed that it was necessary to be in a covenant relationship with God in order to be redeemed from one’s sinful condition, that God had chosen to reveal salvation through preaching, and that the Holy Spirit was the energizing instrument of salvation. Calvinist theology and polity proved to be major influences in the formation of Puritan teachings. This naturally led to the rejection of much that was characteristic of Anglican ritual at the time, these being viewed as “popish idolatry.” In its place the Puritans emphasized preaching that drew on images from scripture and from everyday experience. Still, because of the importance of preaching, the Puritans placed a premium on a learned ministry. The moral and religious earnestness that was characteristic of Puritans was combined with the doctrine of predestinationinherited from Calvinism to produce a “covenant theology,” a sense of themselves as the elect chosen by God to live godly lives both as individuals and as a community.

[...] The New England Puritans fashioned the civil commonwealth according to the framework of the church. Only the elect could vote and rule. When this raised problems for second-generation residents, they adopted the Half-Way Covenant, which permitted baptized, moral, and orthodox persons to share the privileges of church membership. 

Other variations of the Puritan experiment were established in Rhode Island by Roger Williams, who was banished from the Massachusetts Bay colony, and in Pennsylvania by the QuakerWilliam Penn.

-- @Britannica, https://www.britannica.com/topic/Puritanism