Puritan was a nickname first given to those people who wanted to "purify" the church of England because they felt it was corrupt. This group of people, who felt the church was corrupt, left England and went to America in search of religious freedom. They established Puritanism or the "New England Way" of living and it was very strict.
The New England Way was a very rigourous and strict examination of a Puritan's beliefs. This was conducted to identify those people that were qualified for church membership. It was actually designed to limit church membership and changed for the next generation of members. Due to the limited number of members, the half way covenant was created for the next generation.