Kirk's battle of wits with the Romulan commander in "Balance of Terror". For both captains.
Also for the Romulans in general. The Romulan ship is smaller than the Enterprise and fusion-powered, meaning she should be a lot less powerful than the Federation ship... And yet, the Romulan Bird of Prey outguns the Enterprise, with the latter only hope of surviving being hit is to be far away enough it loses power. When you add that the Federation crew isn't too surprised by this, you start to understand exactly why it took the creation of the Federation to take on them and, over a century later, the Federation is still shit-scared of them.
Return of the Archons
Spock decided to cut loose a little and actually punched out a guard.
Once, Spock was under the influence of the Lotus-Eater Machine, and Kirk had to get him mad enough to shake off its effects - not easy with a Vulcan. Once he's done insulting him, his race, and his girlfriend, and Spock's finally had enough, he throws Kirk around like a rag doll, even bending the metal bar Kirk tries to use to keep him at bay. He's about to remove Kirk's head when he finally comes to his senses. A hilarious moment for Kirk when he's trying to provoke him, and a rare look at what a Vulcan who's finally had enough is capable of.
Kirk: It isn't every first officer who gets to belt his captain ... several times.
In many ways, a certain scene in the 2009 movie is a Call Back to this one.
Dr. McCoy when Khan suddenly grabs him and holds a scalpel to his throat:
You do not talk about Moments of Awesome and not mention the Kirk/Khan fight.
Except to note that Kirk finally defeats Khan by hitting him with a plastic toy hammer.
From the same episode, it's a brief see-it-or-miss-it moment, when Scotty punches out one of Khan'ssuperhumanminions in one blow. Then, had enough smarts and speed to escape from the knock-out gas entering into the room.
The Devil in the Dark
On a professional aspect, Dr. McCoy saved an entire species by successfully treating a severely wounded Horta, a silicon-based lifeform - something he was not only completely unfamiliar with, but didn't believe was possible in the first place until that very day.
The Corbomite Manuever
Kirk: This is the Captain of the Enterprise. Our respect for other life forms requires that we give you this... warning. One critical item of information that has never been incorporated into the memory banks of any Earth ship. Since the early years of space exploration, Earth vessels have had incorporated into them a substance known as... corbomite. It is a material and a device which prevents attack on us. If any destructive energy touches our vessel, a reverse reaction of equal strength is created, destroying —
Balok [voice]: You now have two minutes.
Kirk: — destroying the attacker. It may interest you to know that since the initial use of corbomite more than two of our centuries ago, no attacking vessel has survived the attempt. Death has... little meaning to us. If it has none to you then attack us now. We grow annoyed at your foolishness.
This line from Sulu pretty much sums up Spock's brilliance:
Sulu: Try to cross brains with Spock, he'll cut you to pieces every time.
A Taste of Armageddon
A mix of Moment of Awesome and Funny Moments, the episode "A Taste of Armageddon": Kirk and Spock have snuck out of their holding room, and see across the hall a group of armed guards escorting people to the termination booths. Spock casually walks up to one of them, says "Sir, there is a multi-legged creature crawling on your shoulder" gives him a Vulcan nerve pinch, snatches his gun, and calmly walks away while the other guards just stare.
Scotty standing up to an all-too-trusting ambassador in "A Taste of Armageddon".
Ambassador Fox: Diplomacy, gentlemen, should be a job left to diplomats. You will, of course, immediately resume a peaceful status.
Scotty: No, sir — I will not!
Fox: What did you say?
Scotty: I'll not lower the screens; not until the captain tells me to.
Fox: You are taking orders from me.
McCoy: Mr. Fox! They've faked a message from the captain, they've opened fire on us, now you expect us to trust them openly?
Scotty: I understand, sir; but I'll not lower the screens.
Fox: Your refusal has jeopardized the success of this mission! I could have you sent to a penal colony for this; my authority—
Scotty: I know all about your authority, but the screens stay up!
This was supposedly based on a real-life experience James Doohan encountered in the Canadian armed services, according to a documentary included in the first season collection, and the Star Trek wiki:
As a lieutenant in the Royal Canadian Artillery, James Doohan was threatened with court martial for real for saying "No sir, I will not," to a visiting colonel when he realized a training exercise order would entail blowing the heads off some of his own men. Fortunately, his immediate superiors backed him up and, like his fictional character, he was eventually promoted to captain.
Part of the solution to the crisis involves setting things up so that unless Kirk countermands his order, the Enterprise will destroy all life on the planet in two hours. The true awesome comes when you realize that it is not a bluff — the question is whether the Enterprise crew will do it, not if the Enterprisecan do it.
Spock trying to convince T'Pau to release Kirk from participating in the Kal-i-fee. Spock, who has been beaten up over his human half for his whole life, gives his people more fuel and even takes a few shots from T'Pau, the woman Kirk described as "all of Vulcan wrapped up in one package." While Spock was never in any physical danger in this scene, it may be the bravest thing he ever did in the entire series.
Likewise, there are his parting words to T'Pring and Stonn, after 'praising' her for the coldly logical way that she manipulated events to her favor.
In "Amok Time," McCoy pulls a last-minute save out of nowhere to keep Spock from killing Kirk, and bluffs the hell out of everyone there, including Kirk, in order to pull it off. And gets in a dig at Spock at the end for good measure. Beautiful.
In "The Changeling", Kirk uses a Logic Bomb on the "perfect" Nomad probe, revealing that it is actually faulty and thus, by its own logic, must destroy itself. Spock compliments him afterward. When a Vulcan compliments you on your logic, you done good.
Kirk: You must sterilize in case of error?
Nomad: Error is inconsistent with my prime functions. Sterilization is correction.
Kirk: Everything that is in error must be sterilized?
Nomad: There are no exceptions.
Kirk: Nomad, I made an error in creating you.
Nomad: The creation of perfection is no error.
Kirk: I did not create perfection. I created... error.
Nomad: Your data is faulty. I am Nomad. I am perfect.
Kirk: I am the Kirk, the creator?
Nomad: You are the Creator.
Kirk: You are wrong! Jackson Roykirk, your creator, is dead. You have mistaken me for him. You are in error. You did not discover your mistake. You have made two errors. You are flawed and imperfect and you have not corrected by sterilization. You have made three errors!
Spock in Star Trek: The Original Series, or specifically the bearded version in "Mirror, Mirror," when he warns his version of Sulu that if he is killed, he has associates who would avenge him, capped off by appending "...and some of them are Vulcans." Sulu's "Eep!" look at this statement and the accompanying music sting is all you need to convey the fact you do not mess with this Spock!
Uhura intimidates the mirror Sulu into backing down with nothing but a cool voice, a steely stare, and a dagger the size of her hand.
The Doomsday Machine
The final five minutes. A dramatic countdown as Kirk pilots the USS Constellation on a suicide run...but the transporter shuts down before he can beam off, effectively trapping him on the doomed ship, while Scotty races to fix the transporter. One of Trek's all time biggest Moments of Awesome.
When Kirk finds out the mentally unstable Decker had taken control of his ship and is expecting Kirk to follow his orders as well, he refuses and tells Spock to relieve Decker of command, on his personal authority as captain of the Enterprise. Also one for Spock, who proceeds to do so with the threat of being hauled off by security if Decker doesn't go quietly. Decker claims he's bluffing; Spock simply responds, "Vulcans never bluff."
Uhura gave a very convincing performance to the androids, pretending to sabotage the Enterprise crew's attempt to sabotage the androids.
Scotty got another one when he decked a Klingon who had called the Enterprise a garbage scow in "The Trouble with Tribbles", after he stopped Chekhov from attacking them for insulting Captain Kirk.
Kirk: "You hit the Klingons because they insulted the Enterprise, not because they..."
Scotty: "Well, sir, this was a matter of pride!"
"I didn't mean to imply that the Enterprise should haul garbage, what I meant to say is that it should be hauled away (fully opens eyes) AS garbage! *Evil Laugh*"
Those are indeed, fighting words.
Cyrano also deserves mention for his Moment of Awesome in this same scene. He watches the brawl while gleefully consuming multiple glasses of wine (free of charge, thanks to the bartender's absence). He then fairly waltzes across the room, carefully avoiding all the humans and Klingons being thrown around, then stands at the door and raises his wine glass, preparing to drink it. The bartender returns at that moment and swiftly snatches the glass from his hand without even breaking stride. Cyrano, undaunted, pulls another wine glass out of his pocket and drinks it with a self-satisfied smirk. Pure. Win.
The Deadly Years
When Kirk pulls the Corbomite bluff again to escape a Romulan attack, minutes after having recovered from Rapid Aging-onset senility. The Romulans' hasty retreat combines this with Crowning Moment Of Funny.
Kirk: Bones? You could stop all this by saying no. That's why I called you all here together. We'll all be deeply involved. It must be unanimous.
McCoy: Then I'll still want one question answered to my satisfaction. Why? Not a list of possible miracles, but a simple basic understanding of why that overrides all danger! And let's not kid ourselves that there is no potential danger in this.
Kirk: They used to say if man could fly, he'd have wings. But he did fly. He discovered he had to. Do you wish that the first Apollo mission hadn't reached the moon, or that we hadn't gone on to Mars and then to the nearest star? That's like saying you wish that you still operated with scalpels and sewed your patients up with catgut, like your great-great-great-great-grandfather used to. I'm in command. I could order this. But I'm not. Because Doctor McCoy is right... in pointing out the enormous danger potential in any contact with life and intelligence as fantastically advanced as this. But I must point out that the possibilities, the potential for knowledge and advancement, is equally great! Risk... risk is our business! That's what the starship is all about. That's why we're aboard her! You may dissent without prejudice. Do I hear a negative vote?
Spectre of the Gun
Regular Spock had his moments too. This especially applies to "Spectre of the Gun," where the gang is trapped in a surreal Death Trap nightmare of no escape. When Spock reveals he figured a way out, the atmosphere instantly changes to hope as you hang to on every word Spock says as he explains the truth of the situation and the solution, which of course works perfectly in suddenly making the landing party the ones with the real edge in that nightmare.
Kirk and Uhura have the first interracial kiss; mainly because, in that same episode, Kirk acts like a horse so a midget can ride him. And later, the same midget is being telekinetically controlled by some aliens to attack Kirk with a knife. And Kirk responds in kind.
The behind the scenes story is pretty great too. The network was skittish about the kiss, and asked that they shoot a few takes where Kirk and Uhura stop before they actually kiss. William Shatner and Nichelle Nichols responded by deliberately screwing up every single take the director tried to do of them not kissing, forcing the editors to use a take where they did.
In fact, if you read their respective autobiographies, it becomes pretty clear that the events of the Kiss Scene are the only thing the two see eye-to-eye on. Nicols and Shatner really don't get along.
When Dr. Janice Lester, an ex-lover of Kirk's, was in his body and acting as captain and trying to sentence both Kirk (who was in her body) and Spock (who found out about the switch) to death, Scotty and McCoy privately had a conversation about what to do, as they suspect something but have no proof to directly prove it.
Scotty: "Suppose you voted with me in favor of Spock. That's two to one and Spock is free. What do you think the Captain will do?"
McCoy: "I don't know."
Scotty: "You know, all right. It'll stick in his craw. He'll never accept it."
McCoy: "We don't know that."
Scotty: "I tell you, he won't. Then, Doctor, that's the time we move against him. We'll have to take over the ship."
McCoy: "We're talking about a mutiny, Scotty."
Scotty: "Aye. Are you ready for the vote?"
McCoy: "I'm ready for the vote."
Janice Lester gets one for recording the whole conversation above and effectively able to use this to sentence Scotty and McCoy to death for trying to mutiny against her, the "captain".
Chekov and Sulu defying Lester's orders toward the end of the episode. At this point, she has already locked up Kirk, Spock, Scotty, and McCoy and planned to have them executed as mutineers. It would be small beans for her to add their names to the list. But that doesn't matter. These two are not assisting this Ax-Crazy body-swapper in succeeding with her plans.
I think William Shatner deserves a mention for acting as Janice Lester in this episode, being crazy that she was, and throwing a particularly nasty fit once the crew all refused to obey her orders, exposing her as the scheming, insane shell of a woman that she once was.
Also Sandra Smith deserves a mention for doing a fine job of playing a weakened, though still determined and composed Kirk during his time in Lester's body.
A Real Life example, told in Star Trek Memories: Nichelle Nichols had come to work in spite of being in a car wreck on the way there, and fainted on set when the anesthetic took effect. William Shatner pulled his rank as star of the show to take her home before he resumed shooting.