Ship's Master (the captain) Christopher Jones. 
The Master's job is to direct the course of the ship, and to command all the sailors and crew.  Christopher Jones was about 50 years old at the time of the Mayflower's voyage to America in 1620, and he was also a quarter-owner of the ship along with three other men.

Pilots and Master's Mates John Clarke and Robert Coppin.
The Master's Mates were second-in-command underneath the captain; and the pilot's duty was to take charge of the ship when approaching land and guide the ship to a safe place to anchor.  Both John Clarke and Robert Coppin had been to Virginia and New England on previous voyages.  John Clarke had even been captured by the Spanish in Virginia, and held as a prisoner in Havana, Cuba and Madrid, Spain for more than five years before being released.

Cooper John Alden.
The cooper's duty was to build, repair, and maintain the ship's barrels.  The barrels were very important, because they held the food and drink that the passengers and crew needed to survive.  John Alden was about twenty-one years old, and was a distant relative of Christopher Jones.

Ship Surgeon Giles Heale.
The ship's surgeon was exempted from all duties, except caring for the sick and injured crewmembers.  Giles Heale had just recently completed his apprenticeship in London as a Barber-Surgeon, and this voyage may have been his first "real world" job.

Other Crewmembers.
The Master Gunner was responsible for the maintenance and readiness of the ship's guns and cannons.  The ship's carpenter was responsible for fixing any of the wooden and metal fixtures, stopping leaks, and maintaining the good condition of his tools.  The boatswain was responsible for the sails and rigging, and for setting out the anchor.  The four quartermasters were responsible for the cargo hold and the provisions, and for fishing.  The cook was responsible for cooking and providing the daily meals.  The swabber was the person in charge of keeping the ship decks and maps clean.  And there were the ordinary sailors, whose job was to perform many of the laborious tasks, such as hoisting sails, managing day-to-day sailing chores, and steering the ship while at sea.