Will Smith taking an alien ship down using his jet's parachute, which leads to him opening the ship and punching the alien in the face. "Welcome to Earth!"
"Now that's what I call a close encounter."
His badass (and hilarious) explanation of how he's more than willing to repeat that tactic against the aliens:
Capt. Steven Hiller: Wait 'til I get another plane! I'm gonna line ALL your friends up RIGHT beside you!
Russell's crowning moment after realizing that his final missile has jammed, says good bye to his children before flying into the alien ship's primary weapon, gleefully screaming "Hello boys! I'm baaaaaaaaaaack!!" And the aliens promptly learn they fucked with the wrong race.
As well as this, moments before: "In the words of my generation - UP...YOURS!" Most. Badass. Sacrifice. Ever.
The awesomeness of that sequence started as soon as the President shouted "Doesn't anyone have any missiles left?!"
The original script was even better, with Russell in his crop duster with a missile tied on. Note that this means he took off intending to sacrifice himself.
"Must go faster! Must go faster!" —— "Elvis has left the building!" —— "Didn't I promise you fireworks??"
Adam Baldwin gets one too.
General Gray: Is that glass bulletproof?
Major Mitchell: No, sir! (He and several others pull out their guns and start blowing the shit out of the lab window and the alien behind it)
And then Major Mitchell walks up to the alien, all intent and calm, looks it square in the eye, and then delivers the final killing shot. The expression on Major Mitchell's face...
Even more awesome if you imagine that it's John Casey shooting the alien dead. Or Jayne Cobb. Or Kal'Reegar. Let's just say that Adam Baldwin has been gloriously type-cast as a gun-toting badass.
Finally being able to hit the aliens. A moment summed up nicely in one line: Payback's a bitch!
In some theaters, the first missile to penetrate the shields and detonate against the hull elicited an ovation.
President Whitmore's entire Independence Day speech. It is on a great many blogs and lists as one of the top 5-10 (depending on the blogger/writer's scale) greatest movie speeches of all time.
President Whitmore: Perhaps it's fate that today is the Fourth of July, and you will once again be fighting for our freedom. Not from tyranny, oppression, or persecution...but from annihilation. We're fighting for our right to live - to exist. And should we win the day, the Fourth of July will no longer be known as an American holiday, but as the day when the world declared in one voice: We will not go quietly into the night! We will not vanish without a fight! We're going to live on! We're going to survive! Today we celebrate our Independence Day!
The President's firing of the Secretary of Defense:
Albert Nimzicki: I understand you are upset over the death of your wife, but that's no excuse for making another fatal mistake. President Whitmore: No, the only mistake I ever made was to appoint a sniveling little weasel like you as Secretary of Defense! Albert Nimzicki: I don't think you understand — President Whitmore: However, that is one mistake, I am thankful to say, that I don't have to live with. Albert Nimzicki: Mr. President — President Whitmore: Mr. Nimzicki...you're fired.
Will Smith's character again, when he flashes the V Sign, shouts "Peace!" and rams the nuke into the alien mothership's hangar control room.
That entire sequence where they launch their attack; the two decide they aren't getting out of this alive, and light up the cigars before setting off the virus and the nuke one after another, standing up to smile and wave at the aliens for good measure. The kicker is that the alien had no idea what was going on when it saw them. The confused and shocked reaction to the Jolly Roger virus appearing on the screens just sells the whole moment.
Julius telling Nimziki off for chewing David out, reminding him that his son's the only reason they're still breathing.
"Awesome" is still the best word that can describe the scene where people are trying to outrun the ginormous fireballs created by the alien ships. The audience in theaters were screaming for the people to escape. When the sequence ended with the screen's abrupt switch to black immediately followed by the date showing up with the imposing doom sound, the audience in a number of theaters cheered, a true complement in how the film had gripped them to the edge of their seats.
Let's just face it. this alien invasion movie was, is & still a Crowning Moment of Awesome for Roland Emmerich, the actors, and the film itself; not to mention every TV channel aired it mostly every month. ESPECIALLY the Visual Effects, HOLY freakin' SHIT!
This film has held the world record for most miniature models used for effects, and will most likely continue to hold this record for some time, given the increasing amount of CGI effects since 1996.
The part when the aliens blew up the cities was absolutely mind-blowing, jaw-dropping, and amazing.
The White House Destruction is the movie's trope! Literally.
The movie also won an Oscar for VFX, and was nominated for "Best Sound Mixing".
Also worth mentioning is the shot when Boomer leaps to safety, right as fireball passes behind him.
The gigantic fireballs that appear in the sky when the city destroyers are entering Earth's atmosphere.