It is very fitting, with Thanksgiving falling later this week, that the first settlers in colonial New England anchored just off the coast of Massachusetts. The complete story of the Pilgrims often gets lost as the first Thanksgiving seems to dominate the popular understanding of the group. The Pilgrims were a small group of Separatists from Scrooby, England.
Some people (including one so prominent as T. O. Lloyd, among the best historians of the British Empire) call the Separatists Puritans. The Puritans were quite content to stay in the Church of England with certain reservations. The Pilgrims, on the other hand, were not quite so complicit. They decided to break with the church, which was quite the no-no in a very intolerant age in which the civil and religious hierarchies demanded strict conformity. To escape persecution, these Separatists moved to the Dutch city of Leiden. The Netherlands was the most tolerant religious nation in the seventeenth century. However, the sinfulness of the Dutch (as well as their lack of Englishness, which influenced the Pilgrim children) led the Separatists from Scrooby to consider another option--Virginia. The Pilgrims' stay in Leiden is commemorated in a museum, however.
In September 1620, after having returned to England, the Pilgrims began their journey. Their journey was delayed a couple of times because one of the ships that was to bring them over had (apparently debatable) issues over seaworthiness. All of the Pilgrims then crowded upon the Mayflower and embarked on their journey. Leaving in September was not a good idea, as the journey took several weeks. A storm blew the ship off-course, and the small group failed to reach Virginia. Instead, they wound up outside the land that their charter defined, hence the famous Mayflower Compact. They probably didn't land a Plymouth Rock itself, but they nevertheless reached the New World on November 21, 1620. William Bradford was very important, and his diary provides important information on the Plymouth Colony, which lasted until the early 1690s, at which point Massachusetts absorbed Plymouth.
In spite of their popularity because of Thanksgiving, the Pilgrims are much less important than the Puritans who followed them in 1630. The number of Pilgrims was nowhere near as large as that of the Puritans. The Puritans would greatly impact the society of New England with their idea of the godly society, while they basically absorbed the Pilgrims. Nonetheless, November 21, 1620, is a pretty important date in American history. I would argue, however, that it does not establish religious freedom in America. Just because a group wanted freedom for itself does not mean it wants it for everyone else. Hence, they left the libertarian Netherlands. Their journey does indicate the importance of religious belief and how it can impact the decisions and behavior of people, and the beliefs of the Pilgrims and Puritans led them to leave home for their "City upon a Hill."