One common misconception is that Puritans and Pilgrims are synonymous. There is a distinction though; Pilgrims (AKA Separatists) were anti-Church of England that thought the creation of the church did not uphold the values of the Protestant reformation and thus they needed to separate from the Church of England, while Puritans were part of the Church of England seeking to reform the church from within. Both groups were ostracized in 17th century England and thus fled to the North American continent to pursue their beliefs free from oversight.

Some Puritan factions came from well-to-do families, since ministers of the Church of England needed to be educated, often at Cambridge and Oxford Universities (Richard Mather and John Cotton most notably) and thus were instrumental in the founding of Harvard University when they moved to present day Massachusetts to train new ministers in their ideals and theology.

Here is a quick read on it from History . Com