The opening cinematic to the game, combined with Liberi Fatali. It really establishes that you're in for an epic experience.
Squall delivering The Oath, followed by the entire Garden charging into battle despite their wounds and exhaustion.
Made even more awesome by the fact that he had to literally go against his nature to deliver that speech, and that it started as unsure before building up to a really inspirational speech. And then he goes and kicks everyone's ass from Hell to breakfast!
Squall:"......Everybody, this is Squall. How's everyone doing? You're all probably too tired to even stand up after all the fighting. But I want everyone to listen to me... We still have a chance to win, and I need your help. This is going to be our final battle. We're going to attack them before they come in again. To do that, we're going to head straight into their Garden. So I want everyone to prepare for a major collision. Take care of the junior classmen. Irvine, Quistis, Zell, and Selphie will lead the attack into their Garden. As for everyone else, please support them if you can. SeeD was formed to fight the sorceress: at least, that's what I heard. And Garden was created to train SeeDs. So this battle is Garden's destiny and also our destiny. It's a grueling battle, and I'm sure you guys are all exhausted. But I don't want to have any regrets. I don't want anyone to look back and regret this day. So just this once, I want you guys to give everything you got! For yourselves and for me!"
During a cutscene with members of the Garden in a large pitched battle, a Red Shirt clotheslines a Mook off of his motorcycle with the flat of his sword. Easily ranks in the top 10 Red Shirt CMOA's.
Squall's act of insane heroic desperation on the Lunar Station. Rinoa is drifting away into space, running out of oxygen, with no hope left, and Squall, without the slightest hesitation, throws on a space suit and leaps out after her. He has next to no chance of getting to her, he's not even sure if he can help her even if he reaches her, but he still doesn't even pause for an instant. He is going to save her or die trying. That is the definition of love and heroism.
Another example of this in action: the only way into Esthar, which is Squall's only hope of saving Rinoa, is to cross the Horizon Bridge to get to Esthar on foot. The Horizon Bridge spans an ocean, and the trains that run on it are not working. So what does Squall do? He parks Garden at Fisherman's Horizon, waits until no one is looking, and then picks up Rinoa and carries her, on his back, on foot, across the entire bridge. That's a Determinator in action, right there.
Also, Seifer slicing Odin in half by pointing his weapon at him. If that's not badass....
...then what Gilgamesh does to him when he makes his big entrance toting Odin's sword is.
If you're lucky enough to get it (hint: get as many Pinions as you can), being saved from utter annihilation by Phoenix during the Omega Weapon battle is a Crowning Moment of Awesome.
Selphie gets hers when she comes up with a very simple solution to Timber's occupation issues: simply blow up the President's train car with a rocket launcher. Easy and practical!
The Lunatic Pandora storming. How do they get into the nigh-impregnable monstrosity? By charging at it, spamming machine fire, ramming its shields to get the front end of a BFG in, blast a massive hole in the side of the damned thing, then having the impractical landing gear claws grasp onto the side to make a landing. And then the party strolls on out without any mention of how insane it was.
"I'm not a SeeD, but... I can do this... This isn't some kind of game..." Sure, she proceeds to make a pretty rash and foolish move, but this moment is a great showing of Rinoa's determination.
Now, usually when you miss the shot and fail the assassination, you call the mission off. That's not what they teach them in Garden, however. Squall jumps down from the clock tower they fired the shot from, slicing a mook in the process, steals a car and drives through the street filled with panicking crowd and soldiers to the target to take care of business in the old fashioned way. Granted, it fails, but I like to think that the reason none of soldiers who are at max 50 yards away ever intervene during the several minutes Squall fights against Edea and Seifer is that they are too damn scared to go anywhere near the guy (see it by yourself here).
How about Squall saving Zell during the prison escape? A rather shortsighted plan ended up stranding Zell near the top floor of the prison with the rest of the team at the very bottom floor. The team hears some action over the radio and realize they left Zell stranded, all alone, in the middle off a monster and armed guard infested prison that's out to kill them. Zell tries to flee but is caught and seconds away from getting shot point blank in the head when in swoops Squall, cutting the man with the gun to Zell's head down in one shot after running up about a dozen monster infested floors and leaving Quistis and Selphie in his dust. Never threaten one of Squall's friends, he will go to any length to save them.
Not quite as obvious an example at first glance, but the entire first couple of hours of the game are one for Squall and, by extension, SeeD. Squall gets up in the morning, duels his own personal rival before the sun even comes over the hills, and suffers a disfiguring facial scar that also ends with him being knocked unconscious. Then he wakes up and goes to beat the hell out of a local demigod. Then he turns around, goes home, and without even stopping to rest, he hats up and takes part in an amphibious beach assault on a nation that's being invaded, followed by intense urban combat, then dukes it out with a freakish monster-abomination on top of a mountain, and then gets chased down said mountain by an unkillable spider-bot. Then, when he gets back, gets his pips and props, takes part in a huge dance ball, has two women fling themselves at him in their own respective ways, and saves another from another godless abomination of a monster, almost as an afterthought. And this is in one freaking day! Squall pulls off more awesome crap in a single day than most superheroes do in entire movies, and its just another day at the office for him... most of it while he's still a trainee.
Laguna. He's off fighting a war and then decides to drive back to his nation's capital, drink it up with his friends and hit on that woman he's been eyeing for so long.
Later, in Lunatic Pandora, the Laguna team fights off a huge swarm of Esthar soldiers. In between fights he complains about it taking so long it's making him hungry.
Some of the Guardian Force animations, like Diablos, Alexander, and Eden.
The scene where Balamb Garden transforms. Cid said that the place used to be a bomb shelter... who the hell was expecting THAT?
Three teenagers walk into a Galbadian military base with nothing but stolen uniforms and a lot of determination, and proceed to - just as Selphie promised - break everything they can get their hands on, kick the asses of any soldiers who get in their way, cripple a massive ironclad into nonfunctionality, and end by blowing the whole thing sky-high. It's especially awesome if you manage to play it cool and avoid detection until the end, in which case Selphie and her team manage to actively infiltrate the Galbadian chain of command and play everyone but the base commander for a bunch of suckers.
Even better, the speech isn't a case of trying to tell Seifer that he's being an idiot, but rather an attempt to get him to see reason and, when that fails, to tell him that they can't help him fight Squall because they want Squall to win...so they can get their friend back.
Cid, the headmaster, is a kindly old bespectacled man who is being coerced by the real master. When confronting NORG after NORG's failed coup, he blows up and calls him a greedy son of a bitch.
Seifer boasts a claim to fame that no other character in the entire series can make. Odin's One-Hit KOSignature Move, the Zantetsuken, has a horrible hit rate when summoned so every party member can claim they survived it. Seifer didn't just survive or dodge it however, he counters it, bisecting Odin who grinds to a halt as his body splits in two. And to punctuate the defeat, the Japanese characters for Zantetsuken appear as they usually do when Odin appears, then a fourth character joins them — "gaeshi", meaning counter or reversal. Dissidia: Final Fantasy implies this is the only time the Zantetsuken has ever been countered, and claims the incident is still a Berserk Button for Odin.
Meta example: The first time a new player sees Squall perform ''Lion Heart''. Sure, they'll remodel whatever gunblade he had previously into his ultimate weapon. They'll take note of the fact it gives him a new finisher of the same name. But nothing they imagine can compare to the reality.
Speaking of awesome Limit Breaks, Blasting Zone also qualifies. Nothing says "overkill" like a BFS so long that it reaches into space. (Though waiting through a relatively lengthy animation for a single-hit attack that doesn't do much more damage than your normal attack can get a bit old after all.)
It's easily overshadowed by the insane battlefield accomplishments of the party, but consider Rinoa. She's seventeen. She's grown up in comfort and privilege in a well-to-do military family in one of the world's most prosperous nations. And when her country's military forcibly occupies another nation, she has the political awareness and sense of integrity and justice to say 'no, that's not right,' and she picks up and travels to the occupied nation to join a resistance movement, risking her own life to make things better. When her group doesn't have the skills or manpower to accomplish anything, she doesn't give up - she looks to hire specialists who will have the skills to accomplish something, and when she doesn't get anywhere through normal channels she personally travels to their headquarters to make her case to their leader and persuade him to make an exception for her. She may be naive and inexperienced, but there's absolutely no questioning her drive and persistence. Just like Squall, once Rinoa sets her mind to something, she will not stop until she's accomplished it.