The Pilgrims in the 1600s used the Julian Calendar. Ever since 1752, we (the American Colonies) have used the Gregorian Calendar. The reason we switched to the Gregorian calendar was because the Julian calendar slightly miscalculated the exact length of a year by eleven minutes and fourteen seconds. During the 1500s and 1600s, the calendar was off by ten days. Hence, when we look back in time with our Gregorian calendar, we find that the Pilgrim's calendar was ten days behind. So what was September 6 by the Pilgrim's calendar is September 16 by our calendar.
By 1752 the calendar had become 11 days off. So the British Parliament, to fix the problem, declared the adoption of the Gregorian calendar, and renamed September 3, 1752 to September 14, 1752 to fix the eleven-day disparity.
The Parliament also declared that New Years would fall on January 1, instead of March 25. This "New Years" discrepancy is the reason why some dates have been double-dated and look like March 5, 1621/2. This means it was 1621 to them because their New Years had not yet occurred, but it is 1622 to us because our New Years falls on January 1, not March 25.