I'm mostly a singleplayer gamer (yeah yeah, nonsense about the term 'gamer'. Whatever...). I didn't really grow up with games (really started playing games when I was about 14 years old), so I generally don't care much for older games or modern retro-style games. I like games with a good story line and where developers weren't afraid to tell a story. For instance, games doing a 'choose your own ending'-thing like Deus Ex: HR and Mass Effect 3 is a big no-no for me.
But probably the most important aspects of games for me is atmosphere. The gameplay can be shit, or even the story can take a bit of a back seat, but if the atmosphere feels good I'll play the hell out of it. Games like Demon's Souls, Shadow of the Colossus, Half-Life 2, Metro 2033, Okami and many others all have some combination of setting, lore, art direction and music that makes me enjoy them immensely.
And I'm not talking about immersion here. I very rarely feel 'immersed' in a game like I am the character I'm playing, but a game world has to feel in some way lived in or more than just a sequence of levels set up by the developer. Paradoxically, that makes me both like and dislike open world games. I like them because they allow for a world to be built up beyond the constraints like story or 'levels'. But I dislike them as well because open world games tend to lose that attention to detail and flow that linear games can have. The Elder Scrolls games tend to have this problem for me. I love their worlds, but they sometimes forget to be games (I can literally walk around for hours taking in the sights of Skyrim, but I can't be arsed to clear out another damn Draugr cave).
For me, music plays a very important part in building an atmosphere. In films, music plays a large role in building a setting and a mood, and so it is in games as well. A good soundtrack often makes me a lot more interested in a game.