Edward Fuller, his wife, and son Samuel were all on the Mayflower when it arrived in what is now Plymouth, MA. As of this writing there is no reliable source for the name of Edwards wife.
Edward was one of the signers of the Mayflower Compact and when he and his wife passed away his son Samuel was cared for by his brother, also named Samuel, until he came of age.
Fun Fact: Samuel Fuller and Jane Lathrop (Generation 2 below) were married by Captain Miles Standish April 1635.
Generation 1:Edward Fuller and unknown of the Mayflower
Generation 2: Samuel Fuller and Jane Lathrop
Generation 3: John Fuller and Mehitable Rowley
Generation 4: Thankful Fuller and Jabez Crippen
Generation 5: Mehitable Crippen and Bartholomew Heath
Generation 6: Anna Heath and Solomon Knickerbocker
Generation 7: William Knickerbocker and Apaita Parker
Generation 8: William Knickerbocker and Prudence Beadle
Generation 9: William R. Knickerbocker and 1) Catherine Lyon 2)Hellen Burgess
Generation 10: Clarence Knickerbocker and Martha Stuhberg
Generation 11: Clifford, Helen, Eleanor, William, Earl, Luella, Mildred, Donald, Kenneth, and Marjorie Knickerbocker.
Generation 12 and beyond: You and me.
Basically, if you can trace your ancestry to any one of these people then you too are a Mayflower descendant.
Stephen Wainer's Genealogy Weblog 