Protagonist - Doesn't monologue, doesn't think aloud, and keeps his/her cakehole shut in general for over 90% of the game. Not a boring, emotionless iceberg of a character, just not particularly chatty either. (Can't think of any specific game, there's quite a few games with silent protagonists out there)
Story - Takes control away from the player as rarely as humanly possible and makes uncovering most of the backstory entirely optional. Avoids clichés and black-and-white characterizations. (Metroid Prime comes to mind)
Gameplay - Super Metroid. Just Super Metroid.
Musical score - Melodic rock-style soundtracks are really up my alley. (Mega Man X and Super Castlevania IV. I didn't even have to think about this one.)
Setting and atmosphere - Same as the one above the above.
Intro - Doesn't drag on. Gives me a basic idea of what I am and where the Hell I should go. (Again, I can't think of a specific example)
Ending - After a boss battle with a giant alien-brain-mutant-fetus-monster-thing in space, you get greeted by a nice soothing picture of the sea, with a Hitler/Stalin-esque dictator in the background, laughing in his unmentionables atop a hijacked alien spacecraft. Then the credits roll with some nice 24(?)-bit music. (As far as I know, Metal Slug 3 is the only game so far with that kind of ending. Pity, really.)
Online - Don't really care, as long as it isn't completely borked. I'm more into local.