In Star Trek: The Original Series, does James Kirk have any named superior officers, officers who would deliver orders to him and who would expect them to be obeyed? I seem to remember the presence of admirals in the series, but my admittedly skimpy research has failed to unearth any names or character detail.
During TOS, James Kirk holds the rank of Captain and therefore anyone who holds a higher rank is his superior. In later installments the rank of Admiral (and its various sub-ranks, such as Vice Admiral) was established the main rank above Captain, but on TOS the rank of Commodore was more often portrayed as being immediately superior to Captain. During TOS and TAS we meet several Commodores, including:
Commodore Robert April, a previous commander of the Enterprise.
Commodore Matt Decker, killed by the Doomsday Machine.
Commodore Stone, who once oversaw Kirk's court martial.
We also do see a few Admirals:
Vice Admiral Fitzpatrick, who ordered the Enterprise to K-7 during The Trouble with Tribbles.
Following Kirk's promotion to Rear Admiral some time between the end of TAS and the events of Star Trek: TMP, significantly fewer officers outrank him. At least one certainly does, however:
Fleet Admiral Harry Morrow, Commander of Starfleet.
Furthermore, after Kirk was demoted back to captain following the Genesis arc, he is again outranked by Admirals, of which we see several in the later TOS movies:
And of course, throughout all of this, the Federation President, who while not necessarily a member of Starfleet, is Commander In Chief of all Starfleet forces and thus Kirk's ultimate superior.
There may be more, but I'd say that's a healthy start.
There were several officers in the original series who could order Kirk around, and mostly they were Commodores. Commodore Wesley is the one that sticks most in my mind. He commanded the fleet that challenged the M-5 computer in wargames during the episode, "The Ultimate Computer." Just about everything Kirk did in that episode was against his better judgement, under the orders of Commodore Wesley. Another one that comes to mind is Commodore Decker, who seized control of Enterprise to hunt "The Doomsday Machine." Basically, any Commodore in the show had the authority to command or override Kirk, though under normal circumstances Kirk was generally considered in charge of his ship, regardless of the presence of higher-ranking officers.
Kirk normally is under the command of higher ranking officers on starbases and/or at Starfleet Command headquarters on Earth, although that is not usually mentioned in episodes.
Captain's Log, star date 1535.8. UESPA headquarters notified of the mysterious loss of science probe vessel Antares.
In "Balance of Terror":
KIRK: Open a channel to our nearest command base. Quarter hour reports on our position and status.
RAND: We finally received an answer from Command base, sir. They say they'll support whatever decision you have to make.
The Enterprise visits Star Base 11 in "Menagerie", where it is commanded by Commodore Mendez, and in "Court Martial", where it is commanded by Commodore Stone. It is possible that the commanders of the starbase are also the commanders of the region of space the Enterprise is assigned to in those episodes.
In "The Menagerie Part 1" Commodore Mendez aboard the Enterprise gets a message from his superior officer:
UHURA [OC]: Commodore Mendez, urgent. Subspace monitors show Enterprise receiving transmissions from planet Talos Four in violation of Starfleet General Orders.
UHURA [OC]: Captain Kirk is here by relieved. You are ordered to assume command of the Enterprise. Disable vessel if necessary to prevent further contact. Message signed ComSol, Starfleet Command.
The natural assumption is that Comsol is Com Sol, the abbreviation for Commander Sol (Sector?). But once it was possible to make detailed screen captures, it was discovered that Robert L. Comsol decreed General Order 7.
In "Space Seed" Kirk and the Enterprise are apparently assigned to a different sector.
KHAN: I have a few questions first. What is your heading?
KIRK: Our heading is Starbase Twelve, a planet in the Gamma 400 star system. Our command base in this sector. Is that of any use to you?
The star Pollux may be in the same sector, since in "Who Mourns for Adonais?" kirk orders:
KIRK: Lieutenant, relay our position and circumstances to Starbase Twelve immediately.
In "The Alternative Factor" Kirk gets a message:
UHURA: Red Two message in one minute, Captain. Starfleet Command. Code Factor one, sir.
And Kirk talks to Commodore Barstow, who may be the commander of the sector the is assigned to at the moment.
In "This Side of Paradise": Kirk tries to contact higher authority:
KIRK: Lieutenant, put me through to Admiral Komack at Starfleet.
In "Amok Time" Kirk's superior is also Komack:
KIRK [OC]: Lieutenant, get me Admiral Komack at Starfleet Command, Sector Nine. Pipe it down to McCoy's office.
in "The Immunity Syndrome" Kirk gets orders from someone on Starbase 6:
STARBASE [OC]: You will divert immediately to sector three nine J.
KIRK: Sir, the Enterprise just completed an exhausting mission. We're on our way in for R and R. There must be another starship in that sector.
STARBASE [OC]: Negative. This is a rescue priority. We've lost all contact with solar system Gamma Seven-A, which the Intrepid was investigating. And we've just lost contact with the Intrepid. Report progress.
KIRK: Order acknowledged. Kirk out. Mister Kyle, you heard the order. Set course for Gamma Seven-A, warp five.
In "The Trouble With Tribbles" Kirk talks with Admiral Fitzpatrick at Starfleet:
ADMIRAL [on monitor]: Captain, it is not necessary to remind you of the importance to the Federation of Sherman's Planet. The key to our winning of this planet is the grain quadrotriticale. The shipment of it must be protected. Effective immediately, you will render any aid and assistance which Undersecretary Baris may require. The safety of the grain and the project is your responsibility. Starfleet out.
In "For The World is Hollow and I Have Touched the Sky" Kirk has to make the "extraordinary event" of contacting Starfleet Command:
ADMIRAL [on viewscreen]: Perhaps haven't made myself clear. Let me restate it. You have been relieved of all responsibility for the asteroid ship Yonada. Starfleet Command will take care of the situation. (end of conversation)
The end credits name Admiral Westervliet.
In "The Mark of Gideon" Spock requests permission to rescue Kirk from the planet Gideon:
ADMIRAL [on viewscreen]: I sympathize deeply, but Starfleet cannot override Federation directives in this matter.
The final draft script calls him Admiral Fitzpatrick but the cast list calls him Admiral Fitzgerald.
In addition to the admirals and commodores mentioned in the accepted answer, there were also several fleet captains:
The relationship between a fleet captain and a regular captain is not well documented. The distinction may only be important in leading a fleet into battle; normal supervision of captains is ususally done by commodores and admirals.