Nominating one per franchise:
-BioShock (this is including the bathysphere ride - sets up the bare bones, leaves you intrigue, and all while the player is in control)
-Command & Conquer: Red Alert 2 (delicious cheese - maybe 3, but 2 did it first)
-Diablo II ("why did I follow him? Why do things happen as they do in dreams? All I can say is that when he beckoned, I followed. Always...into the East")
-Doom 2016 (as in, everything up to the elevator ride I count as being the intro - in the first few minutes, sells character, premise, and tone excellently)
-Fallout 4 (TBH, I've only played a bit of Fallout 3, but having watched all the intros, I think FO4 actually does the best intro)
-Golden Sun: The Lost Age (it's a simple summary of the previous game, but damn, that music...)
-Halo 2 (in the scope of a few minutes, recaps Halo 1, recontextualizes Halo 1, sets up excellent atmosphere, sets up the direction of the plot, and ends with a nice one liner)
-Killzone 2 (damn, Visari needs to give his speech writer a raise)
-The Legend of Zelda: The Wind Waker (I don't think this is the best intro per se, but since I'm judging this on narrative, it does a good job of summarizing OoT, summing up what happened afterwards, leaving you with a big mystery, and good music to boot)
-Mass Effect ("they called it the greatest discovery in human history. The rest of the galaxy calls it...MASS EFFECT")
-Overwatch (yes, I'm serious. In the scope of a few minutes, the intro cinematic sums up the premise, the gameplay, the characters, and the tone, and even if you only want to watch the cinematic by itself, it's a fun watch)
-Resident Evil REmake (manages to sell the premise, and strike a balance between the cheese of the original, and the more serious nature of the remake)
-StarCraft II: Legacy of the Void (while the best SC intro is for Brood War, if we're judging this based on selling plot, then LotV wins. Also helps that the intro cinematic is actually kickass)
-Star Fox 64 (delicious cheese with a side of plot)
-World of Warcraft: Wrath of the Lich King (while pretty much any Warcraft game from 2 onwards could count, confining this to 1 per franchise, I'm giving the win to WotLK, per its use of music, visual storytelling, and what little dialogue there is is highly effective)