Darth Vader was a man of great importance. He carried out highly sensitive missions for the Emperor, ranging from killing Jedi in the Great Jedi Purge to retrieving the top secret Death Star plans in A New Hope. However, with his appearance and the fact that he doesn't try to hide his affiliation with the Empire, nor that he is a man of importance, he must have had some official role and/or rank within the Empire.
A few other questions have touched on this, for instance Was Vader sincere with Luke in ESB?, Why was Darth Vader called Lord Vader?, How well known were the events of the original trilogy? and Why would the force become so mysterious in just 20 years?, but none of them really answered what I was wondering, and some even raised more questions.
This was my initial question, what was his public role in the Empire? It seems to me that while he may not be present in the capital, or even seen with the Emperor, he should have an official role or some sort of title that the general public can use to identify him with. Calling him 'Lord Vader' could be enough, I guess. The Empire wasn't exactly a transparent ruler, so even if some people (like the rebels) realized that the title of 'Imperial Lord' didn't really cover his actual roles, they wouldn't be able to do anything about it. But did he have an official role in the Empire, and are there any canon/non-canon references that touches on this?
When I thought about this for a while, I realized that the real issue is in fact his...
Rank within the Empire
This is also somewhat related to why the imperials were rude to Vader. The most upvoted comment there suggests that Vader was outside the normal chain of command, which would make sense in A New Hope, but during the assault against the rebel base at Hoth in The Empire Strikes Back, Vader is clearly in charge. Is he a special case when it comes to rank and everyone just sort of obey him, or does the Emperor explicitly tell other officers when Vader is in charge?
If he is outside the chain of command, but still seen as a superior officer and a powerful member of the Emperor's inner circle, how could someone like Motti, Death Star admiral and Chief of the Imperial Navy have the audacity to be disrespectful? I realize that the Force is seen as 'mumbo-jumbo' even by Imperials, but that aside, he must still be regarded as a high-ranking officer. If someone like Motti lacks faith, surely other Imperial officers must feel the same way? Was Motti just stupid enough to let it show?
If a high ranking officer was executing a crucial mission and Vader unexpectedly showed up and wanted to change his plans, would the officer be punished if he refused? Imagine the scenario being played out twice, one where the officer is right, and one where Vader is right. In general, what would be best for the officer to do?
To sum up, what is Vader's official/unofficial role and rank, and how much weight did his words actually carry within the Empire?