What was Darth Vader's official role (or rank) in the Empire?

Question

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.

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Public role

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?

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To sum up, what is Vader's official/unofficial role and rank, and how much weight did his words actually carry within the Empire?

Answer 1

Vader did not hold a named rank within the Empire (at least as of five years after its creation):

Amedda hadn’t said whether he or Vader had command of the mission, and Tarkin was trying to puzzle that out on his own. Vader held an invisible rank.

_{Tarkin, p. 90}

However, he held great authority that was granted directly by the Emperor:

Many believed that the Emperor’s willingness to grant so much authority to such a being heralded the shape of things to come, for it was beyond dispute that Vader was the Empire’s first terror weapon.

_{Tarkin, p. 73}

Thus, while Vader did not have a named rank he was almost always obeyed by Imperial officers and soldiers because of his authority direct from the Emperor. Vader's authority seems to be unspecified, which in practice means he had authority on all matters except those for which the Emperor explicitly specified he did not (e.g. Grand Moff Tarkin was in charge of the Death Star, so Tarkin outranked Vader on the Death Star itself).

Vader also had direct control over various Imperial military assets. He commanded the 501st Legion (which attacked the Jedi Temple with him) as his personal battalion. He also had various flagships, such as the Imperial Star Destroyer Devastator and the Super Star Destroyer Executor (both of which were part of the Death Squadron, the Imperial fleet that attacked the Rebels on Hoth).

Answer 2

He was the right hand of the Emperor

We don't really have an equivalent in America, because the Vice President or Speaker of the House don't have the same encompassing weight... but that's largely due to the structure of a democracy is very diffuse and different from a dictatorship. The closest well-known fake-world example would be the Hand of the King in Game of Thrones - A person close to the king, but not part of the military, or church, or representing a specific area, and technically an advisor to the king, but much more than "just" an advisor.

The closest approximation that we have to something like the Imperial Empire is the Holy Roman Empire, where the political, military, and spiritual leader are one in the same. While the empire had it's normal governmental bureaucracy alongside the military hierarchy, Vader is a third party, the closest real-world example would be a Cardinal Bishop. While he may not have direct linear influence as part of the political or military chain rank-wise, a Cardinal Bishop's words still have the weight of the pope right behind him. This no longer has the weight now that the church is separated from the state (and there are six rather than one), so that no longer has the emphasis from before.

Answer 3

In Star Wars canon, Darth Vader did not have an official rank within the Galactic Empire’s bureaucracy or military. He’s more of a personal aide of high rank to the Emperor who executes or enforces Palpatine’s personal will. He could be seen or called an “Executor-General of the Galactic Emperor” or “Enforcer-General“ alternatively.

In Star Wars Legends, Darth Vader was Supreme Commander of the Imperial Armed Forces effectively making him the equivalent of a Generalissimo, Grand Marshal, or Chief of General Staff, the highest ranks/positions in some state’s armed forces.