Oh man... so many. I'm going to try to keep this short.
Let's start with a classic, the intro to Ninja Gaiden on the NES.
Ninja Gaiden: Intro
This was the first time I ever saw a game do something cinematic with its presentation. It was totally badass.
And I still can't get further than like the 4th level in that game.
Final Fantasy VI: The Opera
The opera scene in Final Fantasy 6, complete with the battle in the rafters during, against a recurring octopus monster that just won't go away, all so the former evil general now turned ally that has infiltrated the opera can sing her song, attract the notorious gambler, and allow you to steal his airship to save the world.
I feel like the previous sentence doesn't make any sense, and I should re-read it to try to make it clear, but fuck it, this train don't stop.
Warcraft 3: Succeeding You
Enough said. But I'll say more, because I like words. As someone who only really got into the warcraft world through Warcraft 3, Arthas was -the- bad guy to me. It's why Frozen Throne was the best expansion in WoW, and why finally defeating the mother fucking Lich King should have been the end of the game. No one cares about Deathwing, or... whatever the hell was going on in Pandaria. Seriously, who was the bad guy in Mists of Pandaria again? Bad feelings?
Bayonetta: Starting a Game Off Like a Bawse
I love Bayonetta. She instantly became one of my favorite video game characters of all time. That's pretty much it, she's awesome, and the intro to her first game was awesome.
Heavy Rain: Scott Shelby is a Chubby Badass
Ok, this isn't technically a cutscene, but all of heavy rain basically is just a cutscene, and one I loved. There's lots of amazing moments in it, most of them spoilers, so I'm just going to share this awesome sequence where Scott Shelby, private investigator, is out for revenge after a mob boss attempted to kill him and his ladyfriend (who probably died in such event, because you're bad at QTEs).
Halo 2: Return to Sender
What a badass way to start a game. Too bad they didn't know how to end one.
Mass Effect 2: Through the Omega 4 Relay
The beginning of the suicide mission. Umph, Mass Effect 2 was so goddamn good. And Mass Effect 3 wasn't. I still have trust issues.
This is probably only part 1