Creating the Firestorm. "I think we've just leveled the playing field," indeed. The entire cutscene is pure awesome.
Equipping said Firestorm with an EMP Cannon or Fusion Lance lets it easily bring down Battleships by itself, even on the hardest difficulties.
Getting an entire squad through a mission without anyone even getting injured. Your NPC advisers will even comment on it.
Earning the Like a Scalpel achievement for the first time. It's a sign of how effective your XCOM operatives have become, able to save the majority of civilians while completely eliminating the aliens without taking any casualties.
The odd pathfinding can lead to some pretty cinematic moments, such as a civilian in a Terror mission jumping through a window to get away from an alien or a XCOM soldier dropping through a skylight to get into a building instead of using the door.
Or deliberately invoking this sort of thing in the name of Dynamic Entry. Have a bunch of aliens cornered in a small building? Have one of your heavies get the door...with a shredder rocket, then have your assault troops rush in with alloy cannons blazing as your snipers mop up the rest before the aliens even know what happened. Single Stroke Battle at its finest.
A quiet one, but the Thin Man you interrogate handles being tortured to death rather awesomely. As opposed to panicking and trying to force his way out like the other species, he'll rather calmly observe his surroundings, give his captors a particularly powerful death glare that works perfectly thanks to the way the camera angle shows his reptilian eyes behind the glasses, and stride right towards the camera as the containment unit seals up.
And then successful interrogation remarks on how the science team was surprised to find how "pliable" the Thin Man was. Either he was unusually cooperative, or their torture techniques got really inventive when they found out about his segmented spine.
Meta-Example: While praised on the internet, X-COM has been a dead game series for the better part of two decades. The majority of gamers in 2012 wouldn't even know it existed, and wouldn't readily embrace it if the game came out today. Add to the mix that the developers constantly praised the original game, and there was a high chance that Enemy Unknown would be the counterpoint to BioShock and Fallout 3, making the die-hard original fans love it and alienating everyone else. And it didn't happen, for the most part. Firaxis managed to take a dead game series, burn off everything that didn't belong, replace it with modern technology and mentalities, and make new people feel what the die-hard fans felt so long ago.
A Sniper with a Plasma Sniper Rifle and Double Tap is concentrated awesome that reaches out and touches the enemy. Having two of them on an open map turns it into a literal shooting gallery as, with the right talents, they can just stand (or hover in the sky with Archangel Armor) at the spawn and are nearly guaranteed to kill or badly wound an enemy or two every turn, without being fired upon in return because they are firing from beyond visual range.
A Sniper with a fully loaded rifle and the Squad Sight 1 (they can target whatever any squaddie can see), Damn Good Ground 2 (additional accuracy and critical bonuses from high ground) and In The Zone 3 (killing an enemy that's not in cover grants a free extra shot) perks. Combine with the Assault's Flush ability 4 forces enemies to leave cover or explosives for lots of dead aliens in one turn.
Alternatively, with careful use of high ground and positioning, a Snapshot 1 (Allows Snipers to shoot post-move with a -20 accuracy penalty) Sniper with In The Zone, Damn Good Ground, and a suit of Ghost Armor or Archangel Armor can snipe an exposed enemy for a free action with ITZ, move into another position, then snipe several now-flanked enemies with more ITZ snapshots, then move again, reload, or take one last potshot at any enemy still in cover and alive.
From the Let's Play series Iskandar's Travels in XCOM EU, a character who, at least in some circles, is becoming a Memetic Badass - or perhaps the Chuck Norris - of XCOM, Colonel Crossblade. Starting out as just one soldier of many, he becomes an Assault, and from there...well, he Assaults his way through aliens like a hot knife through butter. Alien Grenade to the face? You just made him angry. Muton plasma blast to his center of mass? All you did was piss him off. Panicked teammate sending plasma his way? A dodge and a calm report of being under fire. His player normally gave all his soldiers helmets, but as he passed Major he decided Crossblade was badass enough to go without, and gave him the Guile hair. Then comes the psionics lab...and guess who the first soldier out with the gift is? Even when his player decided to try and not have him hog all the XP, using him to shoot to wound a Muton so another soldier can finish it off, Crossblade shoots the alien stone dead. It's reached the point where even the AI seems to recognise how scary he is; in one case a pack of three Mutons all shot at targets further away in better cover, as if afraid of drawing his attention. Writing IC as The Commander, his player states that he's considering retiring him due to the effect on morale his possible loss would cause, but he can't due to his sheer brute competence at killing aliens.
And guess who wound up being the only psyker on the base, and thus the Volunteer? And who ends the Temple Ship mission by running up to the Uber Ethereal and applying plasma directly to its face.
On Council rescue missions, enemies, usually Thin Men, start spawning by dropping onto a building, behind cover, or DIRECTLY BEHIND YOUR GUYS and going into Overwatch. While it never fails to make your heart skip whenever a lanky guy in a suit drops down from the sky behind one of your soldiers, if you're on Overwatch with multiple people, said alien will have about two seconds to realize what a mistake he's made before he's blown apart by two or three guys at the same time.
Final level, Team S.H.I.V (Volunteer and 5 hovertanks). Not even the leader could withstand a barrage of 5 plasma guns bearing down from above.
Killing Sectopods with one volley from a Heavy with HEAT Ammo and a Heavy Plasma.
Or even more awesome, killing one with a single shot from a Plasma Sniper Rifle. Even the aliens seemed surprised.
An Assault using the right perks and Ghost Armor can guarantee that his Alloy Cannon will crit anything it hits. It only takes two crit AC shots to turn a Sectopod into scrap metal, so add Rapid Fire into the mix and the poor machine will never know what hit him.
In the opening mission of the Slingshot DLC, Zhang is telling your squad about the alien tech he stole when he notices a Sectoid behind them. Cue Zhang casually shooting the Sectoid in the head while still continuing his explanation.
Completing that mission on your first go, with zero casualties. Not only will this likely be your first encounter with Thin Men (you usually have more time to prepare for them), but it will also be your first encounter with a Chryssalid (Way before you're adequately equipped to deal with them). Getting through hordes of Thin Men and a Chryssalid with zero casualties this early in the game? Yeah, pretty badass.
Or lobbing in several rockets blind to either kill him outright (circumventing his speech) or weaken him so a sniper can finish him off with one lucky shot.
Doubles as a Crowning Moment of Heartwarming, but any time things go catastrophically wrong and you pull it out of the fire with only one or two troopers left standing. This troper had a single sniper-class New Meat sergeant that hadn't hit a single thing all mission - despite taking shots almost every turn - and had been brought along for the experience (it was supposed to be an easy mission) left standing out of a team of six - his five fellow troopers had been critically wounded or killed in the previous turns. Said sergeant then proceeded to systematically murder every alien on the map with horrifically accurate sniper fire while taking no return hits himself, including one-shotting a charging Berserker at one point. It's worth noting that the five other troopers were all Colonels.
Oh, and that sergeant? He went on to become The Volunteer.
A similar experience occurred while raiding an abductor ship. My squad ended up triggering a huge force of Mutons, Floaters, and a couple of Berserkers by the second turn, who came pouring out of the main bay in a tidal-wave of violence. A few turns later, the entire squad and most of the aliens were dead - save for a single Heavy. Which Heavy? The survivor of the tutorial. He then proceeded to blast apart all of the remaining aliens and, instead of calling off the mission, he stormed the Abductor by himself and killed every remaining alien, including an Ethereal. He, too, would eventually become the Volunteer.
Sometimes, a panicked soldier will shoot, hit and even outright murder an enemy. Bonus points if said enemy was the one responsible for panicking the soldier.
Doubles as heartwarming, but when a soldier gets panicked at the death of a comrade, before turning around and one-shotting an alien who was threatening another.
The Volunteer pulling a Heroic Sacrifice to prevent the Earth from being consumed by a black hole.
Yahtzee mentioned one of his own in his review, relating how most of his troops were trapped in a UFO with one of the more vicious aliens, and the only two with any moves left were a Heavy and Sniper out of range to get inside and cover. So he Took a Third Option - blew a hole in the side of the UFO with his Heavy's rocket-launcher, then Double-Tapped the alien in the tash for the mission completion.
Finding a battleship, the most powerful UFO if the entire game, and shooting it down with your basic fighter, then taking it on with a normal squad with all ballistic weapons, killing all aliens, capturing a and objective target, and losing no troopers, all during the second month.
Capturing your first alien. Not only is it awesome in the sense that you have captured a living alien for all the benefits that provides, and the difficulty in capturing one in the first place, but it is awesome in the narrative as well. The aliens started off their little invasion by abducting humans. Now humanity is returning the favor.
Turning Mind-Controlled aliens into suicide death seekers. One mind controlled Berserker rounded a corner and came face to face with a Cyberdisk patrol. The patrol lowered his health to almost dead. One assisting shot from a nearby Support lowered the Cyberdisk enough that the Berserker could charge and kill the Cyberdisk. The exploding Cyberdisk killed its Drone and the Berserker...on the last turn before the Berserker regained control of its mind.