Honestly when the RP was created the sectors weren't actually planned to be included in the story much themselves at all, it was more just a case of explaining how they got into the house in the character's history. That's why it says "only relevant for history on the island and character background" as throughout most of the RP coming up their positions in the house won't matter. Although that was a bit short-sighted on my part to expect people to know that and I do love how they've been included so far, so I'll give a more detailed description on the workings of sectors now so they can continue to be included :)
Ranks
Leader - The person in charge of a sector. They are responsible for organising the sector, supplying orders to adepts and apprentices of the sector, planning for the sector's future and have the final say in any matters regarding the sector. They are also responsible for relaying information to the higher up members of the Famiglia on their sector's progress and status.
Adept - A high ranking member of a sector. Adepts are responsible for carrying out the more difficult tasks within their sector and are considered to be in command of apprentices whilst the sector's leader is not available. Adepts often advise sector leaders on their decisions and commands.
Apprentice - The lowest ranking members of a sector. They carry out the every-day tasks for their sector whilst gaining the skills needed to progress. Apprentices are expected to follow their leader's orders and report to adepts when their leader is not present.
Sectors
Wands - Responsible for inspections and maintenance of the island. It is their job to keep up to date with all of the island's going ons and ensure there are no problems occurring or looming. Common tasks include carrying out surveys of the townsfolk, performing inspections of important structures and collating the progress of other sectors and the island's progress as a whole.
Coins - Responsible for managing Regalo's finances. Regalo's economy is run by Coin as they handle every aspect of the island's finances. They are responsible for making investments on the island's behalf, dividing up money between sectors, calculating the island's monetary inputs/outputs, finding where savings can be made and providing a bank for the townsfolk.
Swords - Responsible for defence of the island. Sword members can often be found patrolling the streets ensuring the island remains safe. They act as the island's military and police force as well as serving to train guards and apprentices to hone their skills for protecting the island. They operate on behalf of the island.
Alchemists - Responsible for medicine. Alchemists act as the doctors and scientists of the island, often brewing up odd concoctions to heal any ailments inflicted by the townsfolk and experimenting with further alchemy in their spare time. They are responsible for fighting disease, healing the sick and wounded, providing cures to sell/dispense and for any research that needs doing.
Caves - Responsible for supplies. Working closely with Coin, Cave members ensure that the island's supplies and resources remain constant without wasting any money. Cave members are responsible for finding out what supplies are needed, ensuring that they are obtained, and keeping costs low in the process.
Intelligence - Responsible for diplomacy. Intelligence members act as the island's diplomats both internally and externally as they handle all matters of the island's civil diplomacy and any foreign affairs. Common tasks for an Intelligence member include sustaining relations with foreign entities, finding peaceful resolutions for conflicts within the island, acting as a court for those arrested by Sword, and keeping moral of the island high.
@Lilizuki - So as said before, Oda is free to do what she likes unless her sector's head tells her otherwise. She gets a bit more lee-way than apprentices in that she can have a say in what happens, but not turning up at all or directly going against her superior's commands will still result in punishment. Honestly I'm not sure what else you expected considering she's a member of Sword and Sword has a leader, but hopefully the explanations above help to clarify a tad :)
These were written in something of a rush to stop the rising argument that's going on, so if anyone has any questions regarding the information above please post them! It's highly likely I've forgotten to include something whilst trying to make the post quick haha.