Rounding out the archetypal Freudian Trio with The Spock and the McCoy, The Kirk must balance these opposing personalities while being able to take their advice and choose between them (or literally, choose "between them") without being overcome either by emotion or dispassionate logic, representing what in Freudian psychology is called the ego.
Usually, The Kirk is The Captain or a similar leader who needs to be practical rather than emotional or distant. However, it's not impossible for a show to have The McCoy or The Spock as the leader, though they'll have to be far more ideologically flexible than they would otherwise.
They usually share a lot of the traits of the Reasonable Authority Figure, but depending on the slant of the series he might lapse into less than heroic decisions, or end up choosing one of his two friends over the other more often. That said, the burden of deciding what course of action to take can be heavy, while the task of bringing his friends around to accept said decision is complicated as well. At the least, he's mostly immune to Death By Pragmatism. With poor characterization, may become the Standardized Leader.
Examples
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Anime and Manga
Mu La Flaga is The Kirk in Gundam SEED, but interestingly, is placed in a secondary position, with The McCoy as his commanding officer and the Sergeant Rock-ish Spock as an equal. This makes his meadiating efforts difficult to say the least.
Meowth of Team Rocket in Pokemon, although they're all pretty much on equal footing (early in the series, however, Meowth claimed to be the leader of the Trio and this may be why). Meowth typically subverts this by wholeheartedly agreeing with whichever one of his teammates happens to have the same ideas in mind he has.
Films — Live-Action
According to co-writer Roberto Orci, the 2009 Star Trek film maintains this trope, but swaps Kirk and McCoy:
I'm not sure that this works. Luke is more the naïve farm boy (The McCoy) and Han the more sensible, pragmatic one (The Kirk, since he says Kirk-like things such as "Never Tell Me The Odds") but there isn't really a Spock-equivalent. Obi-Wan is less cynical than Han, and Leia abandons the entire Rebel Alliance just to save Han in Episode VI.
Tom Corbett, Space Cadet: Tom is a classic Kirk, settling arguments between his two teammates almost every time they have to do anything, and almost always rightly.
Harry Potter is very much this, with Ron as his McCoy and Hermione as The Spock. Initially played straight, but the dynamic is played with in later books.
Tobias in Animorphs takes this role whenever he's alone with Marco and Ax.
What do you mean? Out of the Animorphs Marco and Ax are the two most logical. Both are occasionally given to bouts of emotion overwhelming logic (understandable seeing as they're both teenagers) but out of the group they're definitely both closer to the Spock end of the scale than the McCoy end.
A better example from Animorphs would be Jake, with Rachel and/or Cassie at the McCoy end, depending on whether the story is about saving things or killing them.
DS9 had Sisko act more like the entire Freudian trio by himself.
Voyager usually had Janeway deliberate this way, with The Doctor and Chakotay advocating the Emotional choice, and Tuvok and Seven the logical (and possibly Straw Vulcan) or efficient one.
Enterprise of course had Archer, with T'Pol and Trip filling in the Spock and McCoy roles, respectively.
The Doctor has fallen into this trope so far, he can no longer be deduced as either an anti-hero or anti-villian. He will give his life (but has not been able to, yet.) to save anyone, any race (minus the Daleks), and planet, or any specific cascade inspired by certain snake haired monsters of Greek mythology from an alien race known to exterminate any other race... You get the idea.
Believe it or not, Jack O'Neill of Stargate SG-1 wasn't really his quartet's Kirk; that honor fell to Samantha Carter and occasionally Daniel.
Which makes Carter a good fit for the new leader in Stargate Atlantis, replacing another Kirk figure, Dr. Weir. The leader of the main team, Sheppard, tends a bit toward the McCoy-ish in his thinking, though he does have the Kirk-ish role in picking the proper course of action out of McKay's stream-of-consciousness TV Genius-ness, Ronon's rashness, and Teyla's more McCoy-ish tendencies.
Sheppard fully ascends to Kirkdom in the last season. Woolsey is The Spock, while Teyla & Ronon are The McCoy.
Mal in Firefly can be pretty emotional and amoral in his own right, but has the virtue (much as he'd deny it) of listening to his crew before making a decision, but being brave enough to take unpopular decisions regardless.
Lee Adama in Battlestar Galactica usually brings the moral clarity to situations where other characters' judgement is blinded by prejudice or fear.
When he isn't blinded by his own prejudice and fear, that is.
Carly in iCarly, the centre of the trio that includes the Spock like Freddie, and the McCoy-type Sam. In iDate Sam & Freddie where Samantha and Freddie start dating, she ends up having to solve all their fights, until the end where it becomes too much and she tells them if they can't stop fighting they shouldn't date at all.
John Crichton from Farscape went from being The McCoy to The Kirk, while D'Argo and Aeryn started out more The Spock and became Kirk-ish as they went along. Pilot is probably the purest Kirk character on the show
Alien X in Ben 10 Alien Force that takes this concept more than a little too literally...
Ben himself is The Kirk alongside Gwen and Kevin, especially in the first season. Strangely, Gwen and Kevin barely fit as the The McCoy and The Spock respectively (both are intelligent and emotional) and can switch the roles depending on the situation. By Ultimate Alien it's hard to peg any of the three as any one of the archetypes, though Ben is definitely the The Spock in the season one finale since he decides Kevin has to be stopped by any means necessary and The McCoy Gwen is trying to save Kevin from himself.
Of course, much of the time by-the-book Blossom and the equally bull-headed and proud "let's just punch them" Buttercup wound up arguing stubbornly while Bubbles was the one who brought them back together...
In the Global Guardians, Guardsman is the The Kirk, even though he's not the team leader. Achilles, the actual team leader, is The Spock, while Arachne and Ultra-Man are The McCoy.
In The Book Of Stories OCT, the Book is this, becoming the balance of Structure and Purpose, which is the literary equivalent of Logic and Emotion.