I don't really care about graphics, which is why I choose to play ps1 games half the time. If I can tell whats going on, and atleast 40% of things look really, really cool/awesome, it's good enough.
Secondly, is concept, games like Pychonuats, Homeworld, and some rpgs come out on top here. If it is an interesting concept (like the pokemon games were when they first started), or is just cool (blast corps, anyone?!?) then it gets a check-mark here.
For sound, as long as theres some good effects, and a stick-in-your head soundtrack (like paper-mario, for example) then it wins here.
Next comes story, if the game has a cool storyline, it's good; which isn't to say I don't play games with no storyline, digimon world three got me hooked and the story line does not exist until 10 hours into the game (at which point it get suddenly really confusing).
Lastly, and probably most importantly, is entertainment. I look for "omgwtfbbq" moments, like getting a 1000ko count in dynasty warriors, grabbing someone from behind in splinter cell, or ripping people in half in God of War. Funnieness counts here, too, like in "destroy all Humans", the first few minutes seem dead-serious, but then the moment the main character shows up, its hirarious from then-on.
Or if , say, my cousin/friend/magazine recomends it, then I'll just try it 'cuase someone whos played it said-so.