Again, from Dictionary.com:

a series of administrative or military ranks, positions, etc., in which each has direct authority over the one immediately below.

Because more than one person is given authority, there has to be some way to know who is in charge at any particular moment. Off the field, we may have the leisure to deliberate and work collaboratively; in the context of an emergency, we probably do not have that luxury: we may still collaborate and deliberate when and where we may, but someone has to make the decision and ensure that the whole structure focuses on the mission.
This is the purpose of ranks in the service and it is the purpose of uniforms, badges, ids, name-tags, etc.: they are emblems of authority. A uniform and particular insignia defines the kind of authority you have been delegated (you are an officer of the Lawrence County Sheriff's Auxiliary), the level of your authority within the hierarchy (your rank and insignia), and your level of training or specific authorities you have been tasked with (your service pins). Again, from our commission:

And I do strictly charge and require those Officers to render such obedience as is due an officer of this position. And this Officer is to observe and follow such orders and directives, from time to time, as may be given by me, or the future Lawrence County Sheriff, or other Superior Officers...

Within the Auxiliary, and within the Sheriff's Office, there is a clear ordering from top to bottom over who is in charge (well, mostly: we'll get to the exceptions). When we work with other agencies, say coordinating on a single incident response, we work under the principles of the Incident Command System (ICS) and unity of command:

The concept by which each person within an organization reports to one and only one designated person. The purpose of unity of command is to ensure unity of effort under one responsible commander for every objective.

and there is still (or should be) a clear hierarchy from the Incident Commander all the way down. Under the principles of ICS, rank and seniority do not define position within the response, so, for instance, a Major in the Auxiliary can be under the orders of a Corporal of police (and, indeed, that exact situation occurs when working under the Sheriff's Office as well) or a volunteer fireman, an EMT, or a rank-and-file CERT volunteer. What is important is that the Incident Commander assigns positions within the structure and each member of the team knows precisely who their superior is from their direct supervisor all the way up the chain-of-command. Teams work together to fulfill assigned objectives. Within self-contained units composed of Auxiliary volunteers, we will generally follow our usual rank structure (...except when we don't, but, again, we'll get to that).
Chain-of-command also defines legal responsibility. We not only need someone to make the decision, we need someone to take responsibility for that decision, and if necessary, someone whose feet get held to the fire if the decision goes wrong. One's position within the structure, and therefore, one's precise responsibilities can change throughout an emergency. In emergencies, as in military action, members of the chain-of-command can become injured or incapacitated. A defined chain-of-command determines who assumes authority and what authority they assume when their superior is suddenly not there. If a team-leader becomes incapacitated, the second-in-command takes charge. If the Communications Officer becomes injured, a specific subordinate will generally take over.

William Sitgreaves Cox and the USS Chesapeake

Under particularly dire circumstances--- and disasters are by definition dire--- a low-ranking officer can suddenly end up in an unexpected position. Take the situation of one William Sitgreaves Cox, who was a Temporary Third Lieutenant (the lowest of the low, below our Junior Grade Lieutenant) in charge of a gun crew on the USS Chesapeake during the War of 1812. In a battle with the HMS Shannon in 1813, every single one of his superiors up to and including the Captain was wounded and incapacitated. He left the deck to carry his wounded captain to the infirmary. At that moment, however, he was in command of the ship and did not know it. By taking Captain Lawrence below decks, he unintentionally abandoned his post. Confusion over command caused the ship to be captured by the British. Lt. Cox was tried, court-martialed, cashiered from the service and lived in disgrace. While this may seem harsh, it was Lieutenant Cox's responsibility to know the chain-of-command and to determine who was in charge before leaving his post. His error was simple but grave.

Patient Abandonment

We do not operate under the same circumstances as Lt. Cox, but we can end up in roughly analagous situations. One such has to do with patient abandonment. When a medical professional or responder (including an Auxiliary volunteer performing first aid or assisting in Search and Rescue) is caring for a victim, they may not simply abandon the patient without handing off responsibility to someone else. Abandonment can result in civil and sometimes criminal penalties. That means that a volunteer may not simply leave a victim and assume that someone is caring for them; responsibility must be clearly handed off. If you are assisting at a treatment area and a second disaster strikes, leaving your superiors incapacitated, it is your responsibility to determine who is in charge, to inform your new superior or to take responsibility yourself until relieved, requesting help if you are unable or unqualified to discharge your new duties.

Formalities of Command Transfer

This is one of the reasons for the formalities of transferring command. Because legal responsibility is involved with chain-of-command, when it changes, it must be formally recognized in some fashion. ICS requires that an incoming Incident Commander must be briefed by the current IC and that the transfer of command must be announced. This same form must be followed at other levels of supervision: when watches change, when an arriving EMT takes responsibility for a volunteer's victim, when a deputy arrives at the accident where you were first-on-scene, etc. If this is not observed, you are no longer in charge, but you may end up still being held responsible for what happens. Briefings may be written or oral as dictated by circumstances. Sometimes the formalities may be very short, including the traditional: "I relieve you, sir,"  "I stand relieved," but they must be observed in all circumstances.
For example, when handing over the care of a victim to an arriving doctor: "Doctor, your patient is Emily Jones, 36. She was injured when she was thrown clear of her car, has severe lacerations, contusions, and a probable head-neck injury. She has been breathing but not responsive, and her pupils are not dilating. We have immobilized her head and neck and stopped the bleeding. We found a Medic Alert bracelet and she is diabetic." This is a short, complete briefing and it also transfers responsibility with the words "your patient". You are making it clear that you are passing care for the victim to someone of greater skill and giving them the information they need to continue care.
You may end up in situations where you are forced to abandon a victim or a post. A good example would be when you hear the bug-out signal during search-and-rescue. At the start of the mission, you will be told that a specific signal (e.g. two long blasts on a whistle) means "get out now" because the area is unsafe. It may mean that a second tornado is coming through, that a structure is about to collapse, or a gas leak has been detected. You may be caring for a victim you cannot move, guarding a supply cache, or performing some other task for which you are legally responsible. Clearly, in that case, you must follow standing orders and abandon your post, possibly locking or securing as you go, however, you must immediately discharge your responsibility by reporting to your superior and you must clearly document the responsibilities you have abandoned (in activity log and debriefing).

For instance, you must immediately report the number, locations, and conditions of victims you have left behind or of critical equipment you have abandoned and obtain some acknowledgement of the fact. You may have received specific instructions in your mission briefing on how to handle evacuation or you may receive instructions when the evacuation is signaled, for example, to move victims who may be moved or to disable/destroy firearms left behind when abandoning a patrol car or field locker. If you do not have specific instructions or there is an unanticipated problem and you have the means to communicate, you must do what you can to clarify your responsibilities: "Zone-2 Command, we have a victim with a probable spinal injury which has not been immobilized. Do you wish us to move the victim, over?" Your superior will then make the call on whether to stay with the victim, take the time to immobilize, move the victim, or leave the victim temporarily.
As an officer, you have to deal with these situations where your orders-of-the-moment conflict with your assigned responsibilities. You have to get clarification if you can and make your best call if you cannot. But, as an officer, you also have to deal with the reciprocal situation of giving orders and assigning responsibilities to those under your command. That means that it is your job to ensure that your mission briefings or your evacuation orders give your subordinates the information they need to make the correct call (or, at least, 'the best call possible under the circumstances'). That is why we have five-paragraph op orders and ICS forms for team assignments, why we practice giving briefings: they give us a structure to ensure that our orders cover the issues they must cover to be carried out effectively, efficiently, and correctly. If you do not give team members instructions on what to do if they bug out, you are responsible for the actions they take under your orders.

Exceptions To the Chain-of-Command
We have mentioned several times that there are exceptions to the strict chain-of-command. These primarily deal with authority under certain defined specialities, such as medical practitioners, Master-Of-Arms, Law Enforcement Officers, and Safety Officers, etc. These are specialists who are given specific and overriding authority within their specialties. In specific cases, those specialties trump any other authority.
For example, the senior, most-experienced medical practioner present may assert authority over a medical issue. If the President of the United States comes down on an incident and tells you to move a patient, and the doctor standing there says, "this patient cannot be moved," then you defer to the doctor, even if the doctor is not your superior, until the situation is sorted out. When the Safety Officer tells your crew to stop activity, you stop, period. On issues of range or weapon safety, you defer to the Master-of-Arms or Range Master, regardless of rank or structure. Within the Auxiliary itself, these specialists may only be overruled on their specialty if their superior officer relieves them of duty and replaces them. If you are that superior officer, you may do so, if, for instance, you know that the patient is sitting on a ticking time bomb and that moving them is the better option even if it cripples them for life, but understand that your actions will be formally reviewed and you must be absolutely certain that you are making the right call. If you are the specialist in that circumstance, you have the same responsibility to only take a stand when it is absolutely necessary to do so and to explain why if circumstances permit. If circumstances allow it in any fashion, the specialist and the officer in charge must work together to come up with an agreeable solution ("OK, the Safety Officer says we can't continue working. Can we obtain appropriate protective equipment and come back? Can we get someone from HazMat to evaluate the risks? How far back do we have to set our perimiter?")
When you encounter conflicts and exercise authority, every effort must be taken to document it so that your decision can be evaluated after the fact. Why did you leave the victim behind in the building? When were you ordered to evacuate? By whom? One of the responsibilities of an officer is to make sure that paperwork is done and records are complete, delegating that task when necessary. Are you making sure that your teams have someone assigned to be a scribe? Have your teams been given appropriate forms? Have you debriefed team members and made sure that an activity log is submitted? Did everyone sign out of the incident before leaving? This is the less-glorious-but-still-important part of having a commission. If something goes wrong, those pieces of paper might be all that is available to try to sort out the problem: Lt. Smith hasn't been seen since yesterday; did they leave the incident without signing out... or are they still somewhere in the field, possibly incapacitated or dead? Suddenly that annoying piece of paper is very important.