I care about the concept which directs the story's aesthetics, characters and plot. I usually like books or movies because they carry out the concept well.
If a story fails, for me, it's because it fails to carry out the concept. If a pulpy story like the ones in Philip K. Dick books--let's say, a story about a psychic kid in a coma who is God and eats souls--had been written a hundred years before the existence of postmodernism, it would have been terrible. Or it might even lack a coherent story: in Gravity's Rainbow, the story (if you can call it that) is a series of fragmented but connected events which start right after a rocket has been launched, which begin to ALMOST make sense just as the book nears its "parabola", and which fall apart as it descends again--ending right before the rocket hits the ground.
The same with characters--usually, good characters help the concept, but if the characters are cardboard thin to serve the concept, that's alright. In Breakfast of Champions, the characters all serve to reinforce the "everyone is robots" epiphany moment of Dwayne Hoover.
I don't know why, but I think aesthetics might be the most important of the three. It might be that the aesthetic permeates the entire work. In Lolita, the actual "story" (I don't think that's the right word) is ABOUT the fate of the exaggerated fantasy which the narrator fabricates. Nabokov said that without the act of individual creation, the world would "rot and stink" like a corpse. So a storytelling style filled with hidden meanings and allusions is essential to establishing the concept. Thomas Pynchon's books, on the other hand, are filled with the same hidden meanings and allusions, but where Nabokov's characters are self-conscious about creating them, Pynchon's characters wonder if they may be looking too hard or going crazy. They're trying to look for the truth BEHIND the meaning, the rotting and stinking world. So the only way the story can be told is with that mood of paranoia.
A final thought: I think I can appreciate things which have concept and nothing else. I'm thinking of Borges's descriptions of fantasy worlds, which are sometimes just that and nothing else. Though maybe an interesting world is most closely related to aesthetics.
I might be terrible at organizing my thoughts.
Edit: I do think I'd be able to enjoy something if it had exceptionally good characters, plot, or aesthetics, even if it didn't have a good concept.