The Calvinists (followers of Calvin) believed in predestination. The "elect" were those who were predestined (elected) to be with Christ in contrast to all the others who were damned. Calvin had a great influence upon John Knox who spread the Calvinistic theology to Scotland, the center of Presbyterian faith.
The Puritans, on the other hand, were those who desired to reform ("to purify") the Anglican church. The pilgrims who came to this country were divided into "separatists" who broke from the church and "puritans" who wanted to reform the church.