Who were the Pilgrims?
A very simplified explanation is that the Pilgrims were separatist Puritans. They believed that the religious reforms they were looking for were not possible in the Anglican Church. Persecuted in England, some of the Pilgrims who would one day sail the Mayflower spent several years in exile in Holland first.
Despite the popular association of the word "Mayflower" with "rich, WASPy snobbery," the Pilgrims who settled Plymouth were, according to Richard Howland Maxwell, mostly working people. It was the Puritans in the more northern colony of Massachusetts Bay, who also wanted to purify the church but were content to stay within it, that were the more educated and monied class.
"William Bradford, the governor whose leadership shaped the Plymouth colony, had been a fustian worker (fustian is a corduroy-like cloth)," wrote Howland Maxwell. "His counterpart in Massachusetts, John Winthrop, was a trained lawyer who had worked in the English government service." ("Pilgrim and Puritan: A Delicate Disinction," by Richard Howland Maxwell. Pilgrim Society Notes, Series Two, March 2003, on Pilgrim Hall Museum web site www.pilgrimhall.org)
The term "Pilgrim" was not used in the 17th century, and the Mayflower passengers, as well as the citizens of Plymouth Colony, were a mix of Separatists and non-Separatists.
How did they come up with the idea of an annual gut-busting feast?
They didn't. To the Pilgrims and Puritans, a day of thanksgiving or a fasting day were jointly called "public days". These days could be held at any time during the year, and featured fasting, praying and reading scripture.
"It may be hard to see a connection between such earnest supplications and our modern Thanksgiving, " wrote New England history writer Eve La Plante in a 2007 Boston Globe article, "but it was that Colonial holiday that America's founders had in mind when they declared national days of thanksgiving."
The 1621 harvest feast memorized by school children did happen however, as Mayflower passenger Edward Winslow and Plymouth governor William Bradford each recorded in contemporary accounts. And as years passed, festivities surrounding an annual day of fall Thanksgiving grew in popularity, especially in New England, where Christmas was considered a holiday rife with Anglican or Catholic superstitions. The modern Thanksgiving feast and national holiday was a creation of Abraham Lincoln in 1863.
What happened to the Pilgrims?
England began restructuring colonial holdings in 1685. Plymouth Colony had its charter revoked, and was combined with the Province of Massachusetts Bay in 1692.
Where are their descendants?
Most people don't refer to being "Pilgrim descendants" as much as they trace their ancestry to either Plymouth Colony citizens or Mayflower passengers. The Mayflower was the first ship to arrive at Plymouth, but three others, the Anne, the Fortune, and the Little James, all brought settlers who became known in colony records as the "First Comers".
Of the 102 passengers on the Mayflower, only about half were alive by the end of the first year. Yet four centuries can add up to a lot of descendants. The Society of Mayflower Descendants in the Commonwealth of Pennsylvania estimates that there are over 10 million descendants just from the 52 Mayflower passengers who had children. The Pilgrim Edward Doty Society has identified more than 92,000 descendants of their Mayflower ancestor over the last 120 years.
Could I be a descendant?
Given the number of Plymouth Colony descendants, it's possible. And it doesn't even matter if your ancestors were separatist Calvinists or not. According to the Mayflower Society, a genealogical society for Mayflower descendants, "Anyone who arrived in Plymouth on Mayflower and survived the initial hardships is now considered a Pilgrim with no distinction being made on the basis of their original purposes for making the voyage." The Society will accept any applicant who can prove his or her descent from any of the Pilgrim names listed on their website, themayflowersociety.com.
The Mayflower Society website provides a list of state societies; these societies handle membership requests and usually provide a historian who can help with your application. The Mayflower Society does offer a preliminary review form, where you are asked to start with the name of your claimed Pilgrim ancestor and list the line of descent which leads to yourself. (In coming up with your actual research, of course, the best route is to always start with yourself and work your way back.)
The Society will compare your claimed lineage to their own historical files and advise on how to proceed. (Note that the California chapter of the Society has its own preliminary review form.) Another alternative is to approach a society dedicated to a specific Plymouth Colony citizen, such as the Elder William Brewster Society or the Pilgrim Henry Samson Kindred Society.
Needless to say, don't take either approach if your research only consists of Grandma, finishing off her second piece of pumpkin pie and glass of wine, mentioning something about a Mayflower or Plymouth Colony ancestor somewhere in the family tree. Knowing the name of your claimed ancestor is a good start, but carefully work your way up through your generations, watching for any other well-known Pilgrim names; your actual link may be to another Pilgrim, or to several.
Societies usually have the first several generations of Pilgrims' descendants documented, but it will take diligent footwork on your behalf to determine if, and how, your lineage links up to the "first comers" who landed at Plymouth. Happy Thanksgiving!