It is a tradition among the descendants of John Tanner that they have ancestors who came to America in 1620 on the Mayflower. It happens that the tradition is correct, but I would guess that few of the descendants could name the passengers and even fewer could explain how they are related. I propose to give some of the background of the Mayflower passengers and more particularly, those from whom the Tanners are descendants.
The Mayflower passengers were not among the first Europeans to visit the American continent. They missed that opportunity either by 128 years or hundreds of years depending on which history you would like to believe. Theirs was not even the first European settlement. The first permanent European settlement in the continental United States was at St. Augustine, Florida in 1565. The Mayflower passengers did not even settle in the first British colony. That honor goes to the settlers on Roanoke Island off the coast of what is now North Carolina in 1585. That settlement failed, but a successful one was established in 1607 at Jamestown. 
So what is the significance of the Mayflower passengers? Other than black hats, Indians and turkeys, what do we really know about these early settlers and why are they significant? Perhaps the answer lies, in part, in the place where they settled, Plymouth Colony in what is now Massachusetts but was then considered the northern part of Virginia. Rather than the temperate coast of North Carolina or the semi-tropic climate of Florida, the Mayflower passengers landed on the northeast coast, a cold and stormy place with rocky soil.
But the story of the Mayflower passengers is vastly more complicated than any present-day apologist or religious detractor would have you believe. We often refer to these original settlers as "Pilgrims" and this designation has taken on a quasi-religious flavor. But, of the 102 original Mayflower passengers, only 41 came to America as religious refugees; the remaining 61 passengers came for purely economic reasons. The "Pilgrims" as all of these passengers have come to be called were English "separatists" who separated themselves from the Church of England and the rule of King James because they found the practices of the Church to be corrupt and in conflict with the Bible. They originally left England for Amsterdam and Leiden where they lived for 12 years before leaving for the New World.
Much of our first-hand information about these early settlers comes from the journal of William Bradford in a book he later published called Of Plimoth Plantation. The manuscript of Bradford's journal was not published until 1856. The book is generally available in its original format with no corrections for spelling or grammar as an ebook.
See Bradford's History Of Plimoth Plantation from the original manuscript. Bradford’s manuscript contains a list of all the Mayflower passengers and subsequently, it is only those who are descendants of those passengers who are eligible to apply for membership in the Mayflower Society (General Society of Mayflower Descendants). On that list are the names of three passengers who have been proven to be ancestors of John Tanner: Francis Cooke, John Cooke and Richard Warren.
Part of the significance of the Mayflower passengers and their settlement at Plymouth Colony is a document they drafted to attempt to avoid conflict between the Pilgrims and their fellow travelers. This document is now known at the Mayflower Compact and is considered one of the first steps taken in the New World that culminated in the drafting of the United States Constitution.