Puritanism refers to a Calvinist party within the Anglican Church, whose primarily desire was to transform that organisation from Episcopalianism to Presbytarianism (this would have made the situation in England, the same as Scotland).
The Puritans advocated a strict, legalistic Calvinism and attacked what they saw as leftover "Romish paganism." They were also strongly infused with millenialism. The term was first used in 1564 and the earliest man to articulate their position was Thomas Cartwright of the University of Cambridge. Some had fled abroad as the Marian exiles during the reign of Catholic restorationist Mary Tudor, Queen of England; the destinations of Geneva, Frankfurt and Strasbourg were popular.
The term "Puritan" is sometimes used in a broader sense to refer to all of the Calvinistic beliefs active during the 16th and 17th centuries in England and it's territories. For instance, the Pilgrim Fathers of Plymouth Colony in North America. The latter were technically Congregationalist Separatists who advocated breaking away from the Anglican Church and setting up their own instead of trying to "reform" it from within. This belief was known early on as the Brownists, named for Robert Browne. The Brownists spent some time in the Netherlands in 1581.
The Puritans became closely associated with the commercial world and supported the English Parliament against the Royal House of Stuart during the English Civil War. With the victory of Oliver Cromwell and the Commonwealth, the morality of their party became the law in England. The confession of the Westminster Assembly is their best known statement of faith. After the monarchy was restored in 1660, with the English Restoration and Anglicanism with it, solidified by the 1662 Uniformity Act, the "Puritans" technically came to an end. They had given up hope of "purifying" the Anglican Church from within and became English Dissenters (like Catholics, outside of the Anglican Church, but in their case they were rarely persecuted to death for their nonconformity).