Kirk. He's trapped in the middle of an asteroid with his companions, and the Enterprise is presumably too far away to rescue them. Then he pulls out his communicator and requests that Spock beam them aboard. As everyone looks at him, completely stunned, he just smirks: "I don't like to lose."
All of this immediately after explaining that he beat the Kobayashi Maru test by hacking it. Whipping out his comm and summoning the Enterprise only justifies his point. Kirk will play your game but he always plays by his rules.
Minor bit, but McCoy is the only one who doesn't look stunned. He SMIRKS. You can read the expression on his face and it clearly says "That's the Jim Kirk I remember." He's spent the better part of the movie worried for Kirk's mental well being, and even though they're still in dire straits, he's obviously pleased that Kirk has taken hold of the reins again.
The initial attack of the Reliant on the Enterprise. The cuts back and forth as Khan has them lock on, Kirk realizing too late what is about to happen and the phasers cutting in, causing massive damage. The reaction of Khan is fantastic as he finally gets to live out what he's been dreaming of for so long.
When Khan first appears on the Enterprise screen, he's wearing the most shit-eating smirk you can imagine. Kirk's reaction is nothing but stunned disbelief, taking a moment to realize it's Khan and can't believe this guy has come back to haunt him.
Khan, triggering the famous Kirk scream: "I've done far worse than kill you. I've hurt you. And I wish to go on hurting you. I shall leave you as you left me, as you left her; marooned for all eternity in the center of a dead planet, buried alive." (Jump Cut to Kirk's communicator) "Buried alive..."
And again, in the Moby-Dick dialogue: "To the last I grapple with thee, from hell's heart I stab at thee... for hate's sake I spit my last breath at thee..."
"Ah, Kirk, my old friend. Do you know the Klingon proverb that tells us revenge is a dish best served cold? It is very cold... in space."
"He TASKS me! He tasks me, and I shall HAVE him. I'll chase him around the moons of Nibia and around the Antares Maelstrom and ROUND PERDITION'S FLAMES before I give him up!" Hell, in fact pretty much anything that Khan says.
Kirk and company strike back with one prefix code, a few wits, and their knowledge of "how things work on a starship" just when Khan thought victory and revenge were firmly in his gloating grasp.
The whole Battle of the Mutara Nebula, from Enterprise's desperate run into the nebula (with awesome Scenery Porn), to the battle itself, and to Enterprise's painfully slow escape from the Genesis Effect, complete with James Horner's perfect Orchestral Bombing. All that positive adrenaline is then quickly made tainted for you for good measure.
Special mention must go to the Enterprise slowly rising up behind the unaware Reliant. First she torpedoes Reliant's own launcher, leaving a husk behind—then she directly phasers Reliant's nacelle, just short of making it completely explode. Then she fires another torpedo straight at the pylon, severing the damaged nacelle and sending backlash throughout the rest of Reliant.note At this point, the level of debris and simple destruction shown simply hadn't been seen in ST before, and it's glorious.
Kirk giving the order to fire on the Reliant. It's a brief scene, only a word long, and he's reveling in the upset he's about to deliver to Khan.
Let's just say it. Spock's Heroic Sacrifice, even knowing how it ends. Roger Ebert says it best: "He makes a choice in Star Trek II that would be made only by a hero, a fool, or a Vulcan." You get a quick shot of his face before he gets up, and you see him make the decision with no fear, no self-pity, only sheer deduction and dedication to his captain and crew. He just calmly walks into a compartment flooded with lethal radiation and tugs open the stream to fix it. Bad. Ass.
And once it's done, his first concern is to make sure it was worth something, then to comfort his best friend as he dies, showing no pain or fear except for breathless, halting speech. Nobody has ever faced danger with such inimitable dignity.
"Mr. Spock, it's been two hours."
During the Mutara Nebula battle, when the Enterprise rose up behind the Reliant, everyone in the theater cheered.
"Sir, the mains are back online."
With hindsight possibly the greatest and most subtle Sequel Hook ever: "I'm sorry, Doctor, I have no time to explain this logically." (cue the mind meld) "Remember."
Spock gets one just by standing up and straightening his uniform. Because if he's going out, he's going out with some goddamn dignity.
"We tried it once your way, Khan, are you game for a rematch? Khan... I'm laughing at the 'superior intellect'."
Nicholas Meyer for being able to try again and again to get Shatner to actually act rather than overact...and in a few instances of allowing Shatner to overact finds instances for it to be perfect. (Such as the previously-mentioned "KHAAAAAN!", in which it's Kirk who's overacting.)
This was even when Kirk and company were trapped beneath the surface of Regula, and Khan could have transported him up to Reliant to do as he pleased with him. Khan had a proverbial sword to Kirk's throat, and Kirk taunts him.
Kirk: You've managed to kill just about everyone else, but like a poor marksman, you keep missing the target!
A small one, and sadly it doesn't stick, but give Chekov credit: he's got balls of steel. When he and Terrell first meet Khan, Chekov openly sneers at Khan, dismissing him as "a criminal," even when it's obvious that Khan could kill them both, or snap his fingers and have his mooks kill them. Chekov doubles down, calling Khan a liar a moment later. Even after he realizes how obsessed Khan has grown, and how much Khan hates Kirk, Chekov still gives Khan a Shut Up, Hannibal!, basically saying to Khan, "You're the one responsible for the fix you're in, asshole." It's only when Khan reveals the earwigs that Chekov starts to panic (and who can blame him).
When one thinks about it, Chekov has quite a few awesome moments. Even when Khan gets ready to Mind Rape Chekov and Terrell with the eels, Chekov doesn't spill the beans about Genesis; he tries to reason with Khan. Then when ordered to kill Kirk, Chekov can't go through with it, and he ends up rejecting the eel's influence. (Terrell deserves credit here, too. He doesn't know Kirk and has no loyalty to him, so he knows he won't be able to resist Khan's influence for long. So he turns the gun on himself rather than commit cold-blooded murder. Really, Khan underestimated everyone at Starfleet.) Finally, after he's been treated, Chekov goes straight to the bridge and asks Kirk if he can help out. Kirk's reply: "Man the weapons console, Mr. Chekov." And it is Chekov who fires the three shots that kick the crap out of both the Reliant and Khan.
This film singlehandedly turned Khan from being just another Villain of the Week in the original series, to being arguably the most respected and remembered individual villain in an nearly 50 year old franchise. That alone should tell someone the impact that he had on the characters in this film. So much so for Khan that his story is reprised in Star Trek Into Darkness, an alternate universe take on Khan's sheer badassery that also forces another Enterprise crew to really think outside of several boxes, with an interesting inversion on the heroic sacrifice to save that particular Enterprise, and a cameo for the last character that saved the "first" Enterprise.
It really says something that when Old Spock is talking about Khan, he describes him as the single most dangerous being the crew of the Enterprise ever encountered. Please keep in mind that they had faced down the Klingon Empire, cloaked Romulan ships, multiple god-entities, alien Nazis, and a weapon that casually blew up entire planets, and this guy was the greatest threat.note It must be noted that the original Khan was just an egotist who got beat down with a pipe by Kirk, not a literal superman. In addition, Khan isn't very smart, either. However, it's his ruthlessness thinking he's smart that makes him dangerous.
Enterprise clearing the spacedock. While a blatant reuse of the visual effects from the first movie, with the addition of James Horner's Crowning Music of Awesome, aptly titled as "Enterprise Clears All Moorings" to mark Star Trek's naval roots, it turned from an awesome scene to doubly awesome one, when it feels like the Enterprise herself is saying "I'm back". Compare the scene from the first movie to this.
Producer Harve Bennett deserves a nod—when he expressed interest in the movie, saying he could produce something more interesting than the first movie (which had been criticized for its slow pace), Paramount execs were concerned with only one thing: keeping the budget as low as possible. Charles Bluhdorn flat out asked Bennett, "Can you do it for less than fucking 45 million [dollars]?" Bennett, who prior to this was a TV producer, boasted that where he came from (his TV producing background), he could do five movies with that total budget. Needless to say, he got the job.