The Puritans were members of a group of radical Protestants which developed in England after the Reformation.
The word Puritan is now applied unevenly to a number of Protestant churches from the late sixteenth century to the early eighteenth century. However, Puritans did not, by and large, use the term for themselves, and the word was always a descriptor of a type of religious innovation, rather than a particular church. The closest analogy in the present day to the meaning of "Puritan" in the 17th century would be "fundamentalist": Puritanism was a movement rather than a denomination.
That said, the single theological movement most consistently self-described by the term "Puritan" was Calvinist and became the Presbyterian Church. The term was used by the group itself mainly in the sixteenth century. By the middle of the seventeenth century the group had become so divided that "Puritan" was most often used by opponents and detractors of the group, rather than by the practitioners themselves. The practitioners knew themselves as members of particular churches or movements, and not by the simple and nebulous term "Puritan."