Title of the image's episode is "A Man's fight". Appropriate due to this being single greatest moment in the series.
Original animated series
Gendo's Establishing Character Moment in Episode 1: When Eva-01 moves to shield Shinji from a large falling ceiling light, and knocks the six-foot steel fixture into the observation window. Gendo doesn't move, doesn't flinch, doesn't even blink as the metal fixture disintegrates on impact less than a foot away from him, and just continues to stare triumphantly at Shinji.
Episode 2: Scary and freaky as it was, the first time Unit 01 goes berserk. At first it was a Curb-Stomp Battle for the Angel, then Unit 01 gets up and not only gets back into the fight, it CHARGES at the Angel, tearing it a new one then surviving the Angel's attempted Taking You with Me.
In Episode 3, Shinji gets one when he just goes into batshit insane, Unstoppable Rage mode and fucking destroys an Angel against his orders.
The fifth Angel has a long range laser that is absolutely lethal. A plan is formulated to have Rei use a shield while Shinji takes the shot. However before he can her shield melts, so she steps in the path to give him more time. The fact Rei has a Olympic level death wish loses nothing. Though this is mainly because she knows ofWe Have Reserves.
The Jet Alone incident. It may just be a filler, but it's still one of Misato's finest moments. She climbs onto the back of basically a giant walking nuclear reactor, goes inside the heavily radiated core in nothing more than a light hazmat suit, and then when the computer fails she tries to physically push the giant control rods back into the reactor. And all without any moaning, just determination.
Shinji gets one in the same episode. Made even better because it's said by the guy who mustn't run away.
Asuka and Shinji start with a combined one in Episode 8 of the original. Basically, Eva Unit-02 is jumping around on battleships and destroyers in order to plug itself in. Then, nearer the end of the fight with Gaghiel, Asuka's yelling at Shinji for being in her lap, calling him a pervert, etc. What does Shinji do in response? He ignores her and keeps fighting the Angel. That's right - Shinji FUCKING Ikari is the pilot who keeps his eye on the ball and his head in the game. The most cowardly pilot in the cast is the only guy who can manage to stay focused on the task at hand, and at least in this case, is able to keep his head in a crisis. Next to his earlier fights against Sachiel and Shamshel, this one makes Shinji look like a certified Badass.
The fight against the 6th Angel was brilliant. Both Shinji and Asuka manage to push their combined sync level to a point that never is quite revealed (but it's general consensus among fans it seems that it's the highest non-400 sync rate encountered in the TV series), forcing the mouth open and allowing two Iowa-class battleships (the Kentucky and Illinois, two ships that in reality were never finished) to blast it apart.
Shinji defying the odds and whatnot when he reaches into the volcano to save Asuka from certain magma-related death in episode 10, despite the risk of him joining her.
The 3 EVA pilots working together (in episode 11) to defeat the Ninth Angel, Matarael, was truly awesome. They're even allowed to enjoy the victory afterwards! Asuka gets double the awesome, of course, because the plan was her idea, and she actually led the others without acting like a pompous know-it-all like she had throughout the episode. Gendo also has one here, as he masterminds and participates in a method to launch the EVAs completely by hand after the base's power is shut down.
Episode 12, where the THREE Evas put their hands up and stop an angel that is essentially commiting a small scale Colony Drop and then stabbing it in the eye.
Ritsuko and Maya get one in episode 13, when they defeat the Eleventh Angel by reprogramming a computer. This being Maya, Ritsuko gets excessively fangirled for her contribution.
How about the the battle against Leliel, the Twelfth Angel? First, Shinji FINALLY starts acting a bit Hot-Blooded - but then, of course, it Goes Horribly Wrong, and we're treated to a monumental Mindscrew. Skip past that, to the part just before NERV's getting ready to drop the N2 mines, the ground suddenly starts trembling, before the streets upheave as if it's an earthquake, piercing through the "shadow" and through the cracks we see this eerie red light, as if a gate to Hell is opening, causing everyone to freak out. Then, Leliel's "body" goes all black, and Unit 01 punches through its "skin" from the inside out and leaping to the ground as the wound tears open and it starts raining blood, now prompting AsukaLangleySohryu to freak the hell out and even Rei to think Oh Crap. Terrifying? Perhaps. Awesome as hell? Definitely.
Asuka: "What the hell? Is my Unit-02 like.... that thing?"
Shinji’s finest moment came in episode 19, against the 14th Angel. Whatever happens with the other Angels, whether it's Yui going berserk and not him, whatever may happen in the other fights, in this one it's all him. He's brave and he's crazy and he just won't quit. And whatever you thought of him before, he demonstrates here that he's a warrior and a hero. And yes, this being Eva, the next several episodes and EoE likewise see our expectations subverted. But just this once, we get to see him go out to do battle, "fangs out and hair on fire," and kicks ass like an action hero and saves the world. Not to mention performing a Dynamic Entry in a Humongous Mecha.
...and then Unit 01 enters its most nightmarish berserk state ever, slices Zeruel like a banana and squishes its face in before eating it. Horrifying, yes, but awesome too.
You really have to give the kid some credit here. The 14th angel is also known as The angel of MIGHT. Prior to this, Rei and Asuka had thrown everything they had at the creature, and hadn't been able to even scratch it (at one point, Rei even resorted to charging it with an N2 missile and detonated it in the angel's face). It was seconds away from vaporizing Misato and the bridge crew when Shinji and Unit 01 smash through a wall and start an all out brawl in the middle of central dogma. He wasn't just the first person damage it either, He beat the shit out of it, and was getting ready to destroy the core. Too bad the batteries died right then.
When supporting character Ryoji Kaji is about to be shot by a SEELE assassin, he just turns around and calmly asks, "What took you so long?"
Misato bitch-slapping Ritsuko for saying it didn't matter if Shinji lived or died definitely counts, and it emphasizes how much she cares about Shinji.
Rei gets a couple of moments late in the series, first against the 15th Angel, Arael. With an angry, determined sneer she defeats by throwing The Lance of Longinus INTO OUTER SPACE and impaling the Angel with it. And let's not forget the fight with the 16th Angel, Armisael, which ends with Rei calmly trapping the Angel inside herself and the Eva and then initiating the self destruct sequence. Sad as hell, but you gotta admit it was awesome.
It is also accepted as one of the most disappointingly confusing endings ever. This makes it a sort of Crowning Moment Of Awesome for Hideaki Anno and Studio Gainax - they created an anime ending that was so damn confusing, it's still hotly debated almost FIFTEEN YEARS AFTER THE SHOW AIRED, and even became the Trope NamerandTrope Codifier for Gainax Ending, in addition to making the series so culturally pervasive and influential in the first place.
One of the most awesome moments was when Unit-00 went haywire and they were forced to shut it down. Rei trapped inside, the team tells Gendo, "The plug is still extremely hot, you can't do anything, you'll burn your own hands off" then Gendo proceeds to run down, and OPEN THE PLUG WITH HIS OWN BARE HANDS! Just to help Rei, no matter how much of a jerkass he is he does care about someone even though Rei is the living clone of his wife. it's still awesome.
End of Evangelion
In End of Eva Shinji is, well... being Shinji, saying he wants to die after Misato saves him. She first tries to reason with him but when that doesn't work she tries another approach: shoving him against a cage and roaring in his face.
Misato gets a grand moment in End of Evangelion when she guns down several well armed JSSDF members with nothing but a pistol, without bothering to take cover, then kicks one against the wall and shoots him in the head and point blank range. Keep in mind, she is not a trained soldier by any means, at least not on the level of the several men she effortlessly killed.
Actually, given that she holds the rank of Major and the billet of Chief of Operations in a paramilitary organization, it is inconceivable that she had not had any training. This in no way invalidates her Moment, however, as such a move normally results in a stupid death, rather than a cool line and a win.
She's mortally wounded in the process anyway.
Asuka in The End of Evangelion, when she took on nine Evas, each with unlimited power and a weapon that could slice through an AT field like it wasn't there, at once, armed only with a progressive knife and three and a half minutes of battery power. And won. Once.
Of course, given that they all turned out to be playing dead, it's pretty likely that they were just toying with her and waiting for her power to run out instead before tearing her to shreads. Sadistic monsters.
The earlier part where she takes on the JSSDF is this as well. Y'know, by tossing a frigate at them. With the JSSDF's weaponry having no effect on her Eva. (After cowering at the bottom of a lake for a while, she realizes that the Eva contains her mother's spirit and snaps out of it. Then we see the frigate slowly rising fromthe surface...)
Say what you will about End of Evangelion, Rei giving Gendou the finger was awesome. In five seconds she goes from being his Apocalypse Maiden to stealing Adam and essentially changed the game to where she had the whole world by the balls. Whether giving the power to Shinji is heartwarming or a wall banger is up for debate.
The manga version has Shinji several moments where he shows honest signs of Bad Ass. The most epic moment being after Toji is killed during the fight with Bardiel which Gendo forced Shinji to do by using the dummy plug to take over Unit 01. He later faces his father and asks him if he has anything to say about it. When Gendo says no, Shinji tries to punch him in the face and is only stopped by Kaji when his fist is literally an inch away from Gendo's face. The moment is so unexpected that Gendo Ikari is thrown off guard and has a look of complete shock on his face.
Rei basically gets to deliver a Shut Up, Hannibal! to Armisael, who had been trying to convince her that she is evil at heart and her existence is worth nothing.
Asuka's first appearance, featuring her beating the crap of a bunch of thugs outside of an arcade, in her civilain dress attire no less!
Misato in Chapter 84 of the manga version. She blows herself up to kill a squad of soldiers.
In such stark contrast to "End of Evangelion", where he sat and moped, Shinji in the manga takes Misato's urging him on to heart and actually is heading to help Asuka in battle. And instead of sitting and whining when he finds his Eva rendered immobile by bakelite, he says "I have to keep my promise to Misato and help Asuka. Mother....Move. MOTHER! MOOOOVE!" And the Eva comes to life. Damn.
And Fridge Brilliance makes it better- Shinji has just saved, oh, maybe two, three minutes of moping around. Two, three minutes in which, in the actual timeline Asuka fucking dies. Do I smell an AU?
And he keeps good on that vow. Even after causing Instrumentality (though for more noble reasons than in End of Evangelion; he does so thinking it's for everyone's own good rather than for his own self-pity), Shinji not only reverses it, but in the manga's ending, we see that the entire human race DOES come back and Shinji and Asuka have a much happier ending this time around. Doubles as a Heartwarming Moment.
When Gendo shot Ritsuko in "End of Evangelion", that was the end of that. In the manga? Ritsuko staggers to her feet afterward while Gendo is distracted with Rei's betrayal and shoots the bastard in the throat! DAMN!
This really deserves some elaboration on why this is so freakin' awesome. Throughout the manga, Gendo has been a borderline psycho who has just revealed that he's really Jumped Off The Slippery Slope in a rant to Shinji about how he's going to destroy the world and become a god, having done multiple morally questionable things (such as forcefully making Shinji indirectly murder his best friend) to accomplish this. Not only do we get the satisfaction of Rei rejecting and betraying him, we get to see Ritsuko shoot him while telling himhe's full of crap. Damn is a freakin understatement!