if you're enlisted, confer with your next highest enlisted person, NCO, LCPO, whatever, then, if that doesn't help anything take it to the officer's direct superior, but remember the golden rule, keep it on the lowest possible level.
Having been an enlisted soldier and an officer I would have to say that there are situations wherein a superior NCO, Staff NCO or an Officer gives a subordinate (Private) an order which the Private disagrees with. I have had this happen and I jumped the chain of command and taken it to the NCO's superior. The NCO got in trouble.
Make sure if you disagree with an order that you are willing to take the heat. Life can get tough if you are singled out by your superior. You have to be extra careful and don't make any mistakes. So make sure it is worth it.
It depends on the order and in some situations you can disobey the order. Especially if you are ordered to kill someone.
If you are ordered to get a 'hair cut' and have the money (pay day) and given adequate time (24 - 48) hours and you still haven't gotten a hair cut. That is disobedience of an order and you will receive company punishment.
NCO's, Staff NCO's and Officer's get legal training on how and what they can do in giving an order. In most cases they are very careful in the wording and presentation of the order and if they are smart they will have witnesses. LOL.
Nothing if it was given to him by his superior. Insubordination can lead to physical punishment, demotion, or even eventually being dishonorably discharged.
Adding to previous statement:
Do what's reasonable...It also depends on the nature of the order and who it came from. Do what's reason "right" or "wrong". If the order involves nothing illegal and it came from YOUR NCOs or your chain of command, then you MUST CARRY OUT THE ORDER.
If it is blatantly illegal or didn't come from your command, then you have grounds to refuse...respectfully. Any sign of disrespect towards higher ranking personnel makes your argument worthless and you will be subject to UCMJ action.
Unless it's something worth fighting for (life/death or career), don't fight it. Just carry out the order anyway and your commander will take the heat if s**t hits the fan.
Trust me. You WILL face serious adversities after refusing an order, so it better be worth it. After you refuse you must always do the right thing, soldier. If your command hates you, they'll exploit EVERY bad decision you make. Furthermore, they will ALWAYS find a way to be right!
Like I said, who gave the order matters a lot as well. For instance, if an E-6 placed over you gives you an order but the company commander tells you otherwise, take the Commander's order! Obviously, the Commander is your chain of command and out ranks the E-6.
However, if your BN Commander gives you a direct order, his orders become first priority. The bigger the rank, the bigger the order.
An adjutant is an assistant, a bird in the genus Leptoptilos, or a lower-ranking officer who assists a higher-ranking officer with administrative affairs.
admiral
The following link will provide an illustration of the high-ranking officer in the Terracotta Army.
The most Senior Ranking Officer regardless of service
The highest ranking officer in the Army is the Army Chief of Staff. The current Army Chief of Staff is General Martin Dempsey.
lieutenant 
The officer would typically report to a higher-ranking officer within their department or chain of command. This higher-ranking officer would have the authority to address any issues regarding who is in charge or "the boss."
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The Attorney General
Lightoller was the second officer on board the Titanic. He was the highest ranking officer to survive the disaster.
A captain's superior is typically called their commanding officer or a higher-ranking officer.
I think you must receive a spartan officer rank for the new ranking system