Helm: steers the ship. Powers thrusters and engines to imprint acceleration to either side, and counters that acceleration when the ship is on course. Also engages warp drive and checks for nearby celestial bodies.
Navigator: plots courses between points in the space, keeps track of known celestial bodies, and keeps track of the ships's position relative to the frame of reference.... namely, the galaxy, or the nearest planetary system. Is responsible for calculating how much time the trip will take through a given course, around a given obstacle and at a given speed.
Weapons: keeps track of the ordnance on board... from small arms to war munitions such as torpedoes or phaser batteries. Is responsible of maintaining and keeping the weapons and weapon systems operational.
Tactical: is responsible of employing a given weapon or a given tactic in combat. Sometimes overrules the input from Helm or Navigation if the ship is to be secured or employed as a weapon in its entirety. Is responsible for planning battle maneuvers and deploying personnel prior to the battle, and assessing the situation of the battle operation during the engagement.
Science: operates the major sensor systems on board, provides input about possible threats and space anomalies, and is responsible of providing major input to Helm, Nav and Tac regarding future maneuvers.
Operations: distributes the resources of a ship evenly throughout the mission, making sure every station has the resources they require for the time span they require, and provides input to Command about the available tools to be used in a given situation. Sometimes works with Engineering and Science to modify specific systems to fit the mission at hand. Can also be in charge of isolated fleet operations, depending on the class of ship (fighter launchings, crew shifts, shuttle ops, transport routes, etc)
Medical: is responsible for keeping the crew healthy and provide inmediate assistance to casualties during combat. Also responsible for operating specific Science systems that might be available during a given mission.
Engineering: keeps the ship in one piece, basically. Is responsible of providing power to the various systems and maintaining the primary power plant and backup systems. Also responsible for deploying damage control teams in battle and assisting any ship or station in distress.
Communications: is the voice and ears of a ship. The comm officer is responsible of recieving and transmitting information between the native ship and any other contacts in the area. Can also track vessels through the presence of comm activity, and can help Command to better plan an operation by providing input about the resources available within whatever range the ship can operate.
Command: responsible of the manpower factor. Executive Officers handle the interactions between crewmembers on a given ship and mission. Sometimes they might be asked to distribute foreign crews to work alongside native crews, according to their capabilites, which they must assess. Captains or Commanders are responsible for responding to orders from Admiralty and Fleet Command, and must decide which available resources to employ from the input given by the other departments.
....I guess that sums it up...
(been playing Artemis lately... man, command is a pain... XD)