Here I'll be describing the various possible positions on a pirate crew. And not just the ones the Straw Hats have.
Primary Positions
These positions are almost essential to have on a pirate ship. Almost all of them are filled by characters of the Straw Hat crew.
Captain
The Captain commands the entire crew as its leader. In the world of One Piece, he or she is usually the strongest member of the crew. This position in the Straw Hats is filled by Monkey D. Luffy.
First Mate
The First Mate is the second-in-command of the pirate crew. He or she usually oversees the work of the other crew members, and answers only to the Captain. Despite never being officially stated as such in the story, Roronoa Zoro is unofficially thought of as the First Mate of the Straw Hats and has been referred to as such by Bartolomeo before.
The Navigator charts the ships path and keeps it on course. Nami is the Navigator of the Straw Hats.
Cook
The Cook prepares the crew's meals and makes sure the kitchen is stocked. Sanji is the Cook of the Straw Hats.
Doctor
The Doctor treats everyone's injuries and handles the sick. Tony Tony Chopper is the Doctor of the Straw Hats.
Gunner
The Gunner mans and maintains the cannons. He or she must be an expert marksman and usually wields a gun or some type of projectile-based weapon. Usopp fills this position on the Straw Hats as the ship's sniper.
Shipwright
The Shipwright fixes and maintains the ship. Originally, Usopp unofficially filled this position by fixing the Going Merry with his carpentry skills, but the position was officially taken up by Franky later in the story.
Ship
The ship is just as much a member of the crew as the people on board, as evidenced with the Going Merry. When a ship is cared for, loved dearly, and treated like a nakama, it can develop it's own sense of self. The type of the ship needs to suit the crew. Larger crews will need larger ships. And they can't all be made of Adam Wood. Adam Wood is insanely rare and impossibly expensive. Generally, only marine ships are made of metal. But a pirate ship can still be made of metal, it's just a bit rare. Few pirates are willing to be compared to the Marines by sailing metal ships.
Secondary Positions
These positions are more like optional positions, as a ship can easily survive without them, or another member of the crew can have abilities that branch into this area, rendering the crew with no need for the official position. They're useful if you want a larger crew, but don't want to create characters with no positions.
Second Mate
The Second Mate is the third-in-command, answering only to the Captain and the First Mate. The position is normally only present on larger crews.
Swordsman
The Swordsman takes care of close-range combat. He or she must be an expert swordsman and usually wields one or more swords of melee weapons. Zoro fills this position on the Straw Hats as the ship's swordsman.
Spy
This position is given to pirates that are able to get information and steal with out being noticed. Spies must be in good body shape that allows them to be acrobatic and fast to run away if spotted.
Helmsman
The Helmsman steers the ship. This position can easily be filled by any of the other crew members, as it is with the Straw Hats, though it is usually Usopp or Franky. But someone with the official position of Helmsman will be much more skillful with steering the ship than a member of the crew with a different official position.
Lookout
The Lookout mans the crow's nest and uses a telescope or binoculars to gaze out at the horizon, ever alert for approaching ships and land masses. With smaller crews like the Straw Hats, this position can be leisurely filled by any member of the crew. But a crew member with the official Lookout position will probably practically live in the crow's nest. He or she will likely heavily personalize the space due to spending so much time there, such as by adding a hammock or sleeping bag, or stocking it with books to read and snacks to eat.
Rigger
The Rigger handles the ship when docking and shoving off. Usually he or she unties the ship from the dock when setting sail, and ties the ship to the dock when docking. The rigger will also raise and lower the sail, and may also raise and lower the anchor. This position is easily filled by any member of the crew, as it is with the Straw Hats.
Quartermaster
The Quartermaster is in charge of overseeing ship preparations and announcing when the ship is ready to set sail. They also assign crew quarters. It helps for them to have skill with weather knowledge, so as to be able to tell if it is really wise to set sail or not. This position is normally only present on larger crews or trade ships.
Blacksmith
The Blacksmith repairs and maintains all of the weapons, armor, and equipment of the crew. They also make new weapons for the crew. Usopp and Franky both unofficially fill this position on the Straw Hats. They even have separate workshops on the Thousand Sunny.
Tailor
The Tailor mends and sews up the damaged clothing of the crew members. Among the Straw Hats, Nami unofficially fills this position, as shown when she mended Luffy's hat after Buggy tore it.
Barber
The Barber is in charge of maintaining the crew's hair.
A bit of a joke position, Cyberweasel89 saw this in a fan fiction and felt it was humorous enough to add to the list.
Scholar
The Scholar is the researcher and general knowledge expert of the crew. He or she will often provide the crew with information on locations and things they come across. It also helps for the scholar to have knowledge of languages, should the crew encounter people who speak a different language. This position is filled on the Straw Hat crew by Nico Robin as the Archaeologist. "Scholar" is just a broader term for her position, and should be used when categorizing characters.
Chronicler
The Chronicler records the crew's adventures in their log book. They normally don't take part in the major activities and goings-on so as to properly chronicle the events. Chroniclers are usually historians who have noticed the various changes and epic events that tend to happen when the pirate crew gets involved, and thus have joined up with them so as to "experience history as it's being made" and record it accurately.
Merchant
The Merchant keeps record of the crew's inventory and spending. It is usually the Merchant who will purchase new supplies, exchange treasure for Beli, trade with the people of every port town, and hand each of the crew members their spending money. This position is unofficially filled by Nami on the Straw Hat crew.
Musician
The Musician plays music to entertain the crew. They are essential on long sea voyages, where pirates will have trouble keeping themselves entertained. It is officially filled on the Straw Hat crew by Brook.
Janitor
The Janitor is in charge of cleaning the ship and making sure it does not get dirty. The ship can easily be cleaned by any member of the crew, so the only reason for this position is to lessen the work load of the other crew members.
Cabin Boy/Girl
The Cabin Boy or Girl performs the grunt work and menial chores on the ship. Considered to be junior or apprentice pirates, they are usually kept busy so as to teach them the lessons of being a pirate. When there is no work to be done, they are usually forced to polish cannonballs.
Pet
Pirates often keep pets, so to speak. Cats are often kept on ships to hunt rats. And dogs are always faithful companions. And despite what some may think, pirate parrots weren't just a myth. Historically, pirates really did keep parrots as pets. The talking birds kept pirates entertained during long voyages. And since they were considered exotic, they often sold for high prices among the wealthy at port towns. This position is considered to be held by Chopper for the Straw Hat pirates although it isn't.
Infantry
Infantry are the members of the crew that have no position. They are usually unskilled in combat and easily taken down by officially positioned members of an enemy crew. As a result, they are usually assigned to stay with the ship guard the rest of the crew's valuables whenever the main crew are out on a venture. They are usually only present on very, very large crews.