I believe you can read the information in the links on the original post ingame, under the Amarr Empire sub-factions.
I'd need a bit more time than I have now for a better response, but the Privy Council article is official. The Emperor article is good for the parts which cite sources, but the rest doesn't, in my opinion, simply make any sense at all. Personally, I would divide the "central government" of the Amarr Empire into several parts.
(1) The Emperor in person/Court Chamberlain - she has power because she is the Empress etc. etc. (the Court Chamberlain is powerful because he is an extension of the Emperor)
(2) The Imperial Household/Emperor Family - governing the lands directly held by the Emperor (the Chamberlain may have a role here, as well, but I'd expect that for the Empire in general, the main function is that people here have ready access to the Empress and can affect her opinions). May also include imperial troops (household forces), even if the bulk of soldiers are actually provided by other Holders.
(3) The Imperial bureaucracy/Chancellor, ministries and services - keeping the imperial machine running. A whole lot of it runs even without an Emperor, but loyal parts can be used to channel quite a deal of power. Unlike the Chamberlain, whose power comes directly from the Emperor, the Chancellor's power comes from being in charge of running things. Militarily, this means both resource-allocating in the Ministry of War and policing via Ministry of Internal Order.
(4) The Emperor as the First Apostle/High Priest - possibly the most pervasive part of the Imperial apparatus, completely ignored in the article. As we know of the Cult of Tetrimon, the Theology Council was at first the Emperor's clergy. If the Big T is controlled by forces loyal to the Empress, then no matter if a fief is held by a Holder of questionable loyalty, the Emperor is there via her priests.
(5) The Imperial military. The Amarr Navy is an Imperial Navy, directly loyal to the Empress. Of course, different factions have their men and women in the Navy, too, so complete loyalty isn't guaranteed (especially if the Emperor lets her guard down). The Sarum are strong here, and the Amarr Navy article directly suggests that many people fear that the Empress will emphasize the role of the Navy (which may be the most loyal part of the Imperial machine at the moment).
(6) The feudal system of imperial Holders, the Heirs and their Holders. Which might, at least on paper, own the most of the things in the Empire, even if said things are practically controlled jointly by the Holders' staff and the imperial bureaucracy which, in turn, is itself influenced by several feudal factions - and the further you get from the Emperor, the more powerful the feudal component becomes, so Amarr might be 99% run by the Imperial bureaucracy, whereas in the middle of nowhere the show is run by the Holders, unless there are direct imperial/Emperor Family holdings.
The 24th Imperial Crusade, as it is ingame, is completely a CONCORD artifact and is a part of the Empire only symbolically. (Actual 24th would likely - in the case of an all-out war - have huge effect on the society that cannot be coined into a corp.)