Data (and probably the rest of the crew, but mainly Data), when they finally defeat the Klingon ship, Data yells out "YES! UGGH!" with accompanying fist in the air.
Throughout the battle, the Klingons clearly revel in their advantage - making it that much sweeter when Data exploits a little technobable to trigger their cloaking device and force their shields to drop. After the Klingons have their Oh Crap moment, the movie cuts back to Riker.
Kirk and Picard's conversation in the Nexus, where Picard talks the old captain into one last mission, and Kirk offers career advice. The galaxy's two greatest heroes sharing the screen is quite a sight to behold.
It was no accident that this scene was the clip most widely seen in ads, interviews and reviews at release.
The opening sequence with the Enterprise-B attempting the rescue of the transport ships caught in the Nexus. Not only packing enough action and suspense in 15 minutes than most movies do in 90, but also packing Character Development for Kirk in the form of angst that he is no longer THE Captain of the Enterprise.
And at the point when Harriman volunteers to go adjust the deflector dish, Kirk sits down in the Captain's Chair, rubbing his hands on the seat, when he realizes he has to let it go...
Bridge Drop or no Bridge Drop, the fact remains that Kirk died as he lived, a hero to the last.
The fandom will probably never let her forget it, but the fact that Deanna Troi was at the helm when the Enterprise saucer went down, not in a safe and planned descent but a sudden jolt knocking them out of space and down to the planet below, something that was untested and probably considered something of a worst case scenario, and made through with only light casualties (an exact number isn't given, but the novelization said seventeen, out of a crew of over one thousand) is a pretty impressive feat.
Speaking of, the crash of the Enterprise saucer is a Visual Effect Of Awesome when you remember that it was done entirely with physical models, not CGI. She may have gone before her time, but the Enterprise's final flight was quite the sight to see.
Captain Kirk and Captain Picard working together to defeat Soran. There is just something so damn cool about seeing two of Star Trek's most famous characters fighting side by side against a common foe. Also doesn't hurt that Kirk, now getting well along in years, can still best someone in a fistfight.
Soran: (walks along the walkway as before, and is now confronted by Kirk) Just who the hell are you?
Picard: He's James T. Kirk. Don't you read history?
It's gets even better, in the original ending. Soran was actually supposed to beat Kirk in there fist fight and kill him with a cheap shot. Test audiences didn't like that at all. So the script was re-written at the last minute to include the more awesome Dropped a Bridge on Him scene instead.
In the final scene, after Picard gives a short monologue on the importance of living life to the fullest:
Picard: After all, Number One, we're only mortal.
Riker: Speak for yourself, Sir. I plan to live forever.