Auron has tremendous awesomeness, but nothing defines his character like him standing still in an arena where a group of monsters are attacking while a panicked crowd run past him, making a hmph sounds as a huge dragon-like monster roars at him, then killing it with one attack after the FMV ends. Watch the FMV scene. Keep in mind, this was really his introduction, and yet he doesn't disappoint the rest of the way.
The scene where Yuna, after a lifetime of preparation for becoming a summoner and defeating Sin, even being willing to die in order to give the people of Spira a temporary relief from sorrow, find out the truth about The Final Summoning used to defeat Sin in the past and promptly say "Screw Destiny, we're going to end the Vicious Cycle permanently, not just start it over" was her Crowning Moment Of Awesome. The rest of the party quickly follow suit, but it was her moment.
Yuna: No one. I would have gladly died. I live for the people of Spira, and would have gladly died for them. But no more! The Final Summoning...is a false tradition that should be thrown away.
Yunalesca: No. It is our only hope. Your father sacrificed himself to give that hope to the people. So they would forget sorrow.
Yuna: Wrong. My father... My father wanted...to make Spira's sorrow go away. Not just cover it up with lies!
Yunalesca: Sorrow cannot be abolished. It is meaningless to try.
Yuna: My father...I loved him. So I...I will live with my sorrow, I will live my own life! I will defeat sorrow, in his place. I will stand my ground and be strong. I don't know when it'll be but someday, I will conquer it. And I will do it without...false hope.
And there's Auron's lines, of course.
Auron: "This is it! Now is the time to choose! Die and be free of pain! Or live and fight your sorrow! Now is the time to shape your stories! Your fate is in your hands!"
"Hah! I'd never forgive myself—no way! Not if I ran away now. Even in death, ya!"
That whole scene was a gigantic Crowning Moment, really. For your viewing pleasure, check it out.
When the party interrupts the marriage between Seymour and Yuna... with an airship that's currently going down in smoke. Need I mention the chain-surfing? Or the giant guardian dragon? Doubles up with Crowning Music of Awesome.
How about Yuna jumping off the building and summoning Valefor in midair? That freaking rocked!!!
This troper thought it was a subtle awesome moment when, before Yuna jumps, Seymour tells her "If you fall, you'll die!", Yuna simply wipes her mouth with her hand, unfazed - as if to say his forceful kiss didn't break her.
Complete the game once and start a new one. When you first see Auron again looking at Sin's gradual approach and says "Are you Sure?" you'll get goosebumps when you realise that Auron was talking to Sin/Jecht!
Operation Mi'ihen was a CMOA for Sin. I know it's probably considered bad form to give props to an Eldritch AbominationSpace Whale, but look at what that thing did. This troper watched the whole thing, and at the end thought "Oh Crap. My party is so out of their league."
Lulu's reaction to when Tidus says "You're more my type, Lulu."
Lulu: Interesting... I suppose I could consider adding you to my list. I wish you good luck, little boy. You're going to need it.
YMMV, but when Seymour changes into Seymour Flux on Mount Gagazet, his speech to Kihmari and his transformation changes him from a soft-spoken wannabe to a AAA badass. Not to mention actually defeating him is a Crowning Moment for the player.
Seymour's introduction via Authority Equals Asskicking. In the middle of the stadium being swarmed by fiends, Seymour calmly walks onto the dais, makes the Yevon prayer, and summons Anima, and promptly slaughers all of the fiends in the area in less than a minute, all without harming a single human being.
He gets another one when you fight him for the first time, cementing himself as a Magnificent Bastard when you choose to have Tidus talk to him:
In a game with the first voice acting venture in Final Fantasy history, the production team could have easily given Tidus a rigid name and made life easier for themselves. Not only did they let you name him, but they wrote the script in such a way that the gameplay doesn't suffer for it. They had to bend over backwards and push Hey, You! to its limits, but they did it.
Lulu's line the first time she casts Flare. This troper was only eleven the first time she heard it and nearly keeled over from the awesomeness.
Lulu: KNOW PAIN!
Another one for Lulu; "A day with you is never dull!" - followed by her suddenly diving off the side of the airship.
Nearly every character got a CMOA line if they're the ones to speak up at that point.
Kimahri Ronso, stepping up to Biran and Yenke solo, and winning. Biran being the Ronso who, apart from towering over him, is the reason he's hornless. Imagine the pride he hid after the fact. Kimahri's honor was finally restored after ten long years. the leading up lines were also badass.
Kimahri: "Then I must prove my strength!" "This time, I win, I will win."
Here's one for the player: Obtaining the Mark of Conquest. It's a key item you get for defeating Nemesis in the Monster Arena.
Yuna gets a subtle one when Dona insults her for having so many guardians and brags that she requires only one. She calmly responds that she only has as many guardians as there are people she can trust with her life. When you later realize this is considered the common definition of a guardian in Spira (Seymour's talk about "The Code of the Guardian", for example), then Dona having only one becomes something she shouldn't be bragging about: Given her abrasive way of dealing with others, she's probably alienated so many people that she only has one person left she can trust. In other words, not only did Yuna calmly counter Dona's insult, she threw it back at Dona without her realizing.
Tidus in the later half of the game after Character Development. He finds out that he's just a dream of the Fayth and if Yu Yevon, who keeps Sin alive is destroyed, the dream will end and Tidus will effectively die. Tidus accepts this without hesitation and goes ahead with destroying Yu Yevon and right before Yu Yevon is fought, he tells the party that he'll disappear and right before the battle starts, he says "I know it's selfish, but this is my story!" and process to kick Yu Yevon's ass.
Oh, you were going to use an Aeon on Seymour Natus or Seymour Flux? Too bad. The way he taunts you before busting out his Aeon-exclusive One-Hit Kill is what sells it: