The Emperor, although unseen and only mentioned in passing in A New Hope, is in overall command of the Empire:

Tarkin: The Imperial Senate will no longer be of any concern to us. I have just received word that the Emperor has dissolved the council permanently.

Given the events of Attack of the Clones/Revenge of the Sith, it seems likely that by this time the Senate was at best an advisory body and probably in reality was there to provide the illusion of democracy. My take on Vader is that he exists outside both the political and military hierarchies (though probably in the line of succession - see below) and acts as a troubleshooter/enforcer/gofer for the Emperor (along with, from the EU, Mara Jade, Grand Admiral Thrawn etc).

The exact hierarchy between Tarkin and Vader isn't made clear from the film - as HorusKol says, they appear to be equals. The EU suggests that whilst Tarkin was in charge of the entire Death Star project, Vader was aboard as the personal representative of the Emperor - most probably to stop Tarkin getting the idea of staging a coup, using the Death Star to assassinate the Emperor. I mostly think of Vader as existing outside both the political and military hierarchies

In terms of the chain of command, the suggestion is that below the Emperor power is held by the politicians

Tarkin: The regional governors now have direct control over their territories.

but there's nothing said about whether they were reporting directly to the Emperor or whether there was a layer of 'middle management'. The military, in the normal course of events, are subordinate to the political hierarchy.

The whole 'chain of command' question raises the question of what would have happened had the Rebels failed and the Emperor not died aboard the second Death Star. The Sith Rule of Two (one to embody power, one to crave it) implies that eventually Vader would have staged his own coup to depose the Emperor (and in fact he more or less tells Luke on Bespin that this is his plan).

Suppose the Emperor had died of natural causes or at least in a non-violent manner - it seems likely that Vader would have been his chosen successor, but after that I think the line of succession breaks down. In the EU when the Empire is left without the Emperor and Vader, it falls apart into a collection of pocket empires and territories held by warlords using military force.

The Emperor, although unseen and only mentioned in passing in A New Hope, is in overall command of the Empire:

Tarkin: The Imperial Senate will no longer be of any concern to us. I have just received word that the Emperor has dissolved the council permanently.

Given the events of Attack of the Clones/Revenge of the Sith, it seems likely that by this time the Senate was at best an advisory body and probably in reality was there to provide the illusion of democracy. 

My take on Vader's position is that he exists outside both the political and military hierarchies (though probably in the line of succession - see below) and acts as a troubleshooter/enforcer/gofer for the Emperor (along with, from the EU, Mara Jade, Grand Admiral Thrawn etc).

The exact hierarchy between Tarkin and Vader isn't made clear from the film - as HorusKol says, they appear to be equals. The EU suggests that whilst Tarkin was in charge of the entire Death Star project, Vader was aboard as the personal representative of the Emperor - most probably to stop Tarkin getting the idea of staging a coup, using the Death Star to assassinate the Emperor.

In terms of the chain of command, the suggestion is that below the Emperor power is held by the politicians

Tarkin: The regional governors now have direct control over their territories.

but there's nothing said about whether they were reporting directly to the Emperor or whether there was a layer of 'middle management'. The military, in the normal course of events, are subordinate to the political hierarchy.

The whole 'chain of command' question raises the question of what would have happened had the Rebels failed and the Emperor not died aboard the second Death Star. The Sith Rule of Two (one to embody power, one to crave it) implies that eventually Vader would have staged his own coup to depose the Emperor (and in fact he more or less tells Luke on Bespin that this is his plan).

Suppose the Emperor had died of natural causes or at least in a non-violent manner - it seems likely that Vader would have been his chosen successor, but after that I think the line of succession breaks down. In the EU when the Empire is left without the Emperor and Vader, it falls apart into a collection of pocket empires and territories held by warlords using military force.