Given that I could go on for paragraphs describing my favourite scenes, I thought I would make it hard for myself and limit myself to the ONE scene that really 'makes' the movie for me.
The Motion Picture: The Meld - the whole scene is magnificent, but when V'Ger explodes/ascends/evolves and the music winds down and the Enterprise slowly emerges, and the ST theme winds up again. In my mind, the movie is basically saying "THIS. IS. STAR TREK."
The Wrath of Khan: Spock's Funeral - Kirk says: "Of all the souls I have encountered in my travels....his was the most.....human." And then the torpedo slowly rolls out, with Scotty playing Amazing Grace. It's so powerful to this day - and then when the torpedo fires and the they switch to the sweeping orchestral music. My god - great stuff.
The Search for Spock: Stealing the Enterprise - now, I am not a big fan of this movie, but this scene is probably the best 8 minutes of cinema Star Trek ever.
The Voyage Home: There are many great scenes, but for me, it's McCoy right after Kirk decided to do go back in time. "What you're suggesting is we travel backwards in time to capture whales, move forwards in time, drop 'em off and hope the hell this probe knows what to go do with itself." Kirk: "That's the general idea" McCoy: "That's crazy!" Kirk: "You got a better idea now's the time."
I love this interchange because very early in the movie, it seems that the filmmakers and actors knew this movie was going to be ridiculous - so they state that is IS indeed crazy, and now well, we can't argue with them can we??
The Final Frontier: Kirk's tirade upon refusing Sybok's exploration of his pain. "I DON'T WANT MY PAIN TAKEN AWAY I NEED MY PAIN". This entire scene is excellent, especially the sequence with McCoy euthanising his father. TFF should've had more of this and less God creature shooting lightning.
The Undiscovered Country: Sulu - "Target that explosion and FIRE." Takei's delivery makes this one of the most badass lines in all of Star Trek. Move over Captain Robau!
Generations: Picard's Christmas in the Nexus. It's really quite moving to see Picard together with a family, and when he hugs Rene, I have great difficulty not crying.
First Contact: The first contact meeting between Cochrane and the Vulcans. "Live Long and Prosper". Cochrane: "Thanks." For me the first contact storyline IS what First Contact is about. The Borg I can take or leave in that movie.
Insurrection: Geordie watching the sun rise with his natural eyes restored. This movie is very underrated and for me, and is strengthened by poignant little moments like this.
Nemesis: Well....this movie sucks, but at least it has an awesome space battle right??? The Enterprise ramming the Scimitar is pretty cool.
Star Trek 2009: The entire prologue sequence involving the Kelvin, specifically George Kirk slamming the Kelvin into the Narada whilst saying goodbye to his wife and son. It's quite emotionally gripping and sets up the rest of the film superbly.