Let's put it this way. All Puritans were Calvinists, but not all Calvinists were Puritans. What this means is that the term Puritan only really has meaning in the context of English—and therefore, American—history.
On the European continent, the term did not apply; here, there were just Calvinists, members of a number of Reformed Protestant churches. As these Calvinists had successfully built their own churches, there was no sense in which they were purifying the practices of any existing church, which is how the word Puritan originated.
In England, however, it was a different story. There, the Puritans sought to reform the established Church of England from within, purifying its rituals and doctrines according to a strict interpretation of Scripture. In other words, they wanted to make the Church of England more like the Reformed churches in places like Switzerland, the birth place of Calvinism.
For a while, the Puritans were successful, and the Church of England gradually became more recognizably Calvinist in its liturgy and form of government. However, after the Restoration of the monarchy in 1660, they were subject to persecution and intolerance on the part of the authorities, which forced many of them to seek sanctuary abroad in the Calvinist Netherlands and in America, which was already earning the reputation of a haven for those fleeing religious persecution in Europe.