Gameplay!Graphics aren't that important,NES graphics aren't good,but I still like to play them.When I first played Super Mario Bros.(NES),I didn't know what to do,only that I had to save a princess,but I still like to play it,so story isn't that important to me.
Well, gameplay is important - I have a major quirk about having to control the character directly in every aspect possible. I just can't take playing games where combat is resorted to having to choose actions from a list. The #1 reason I never got into Final Fantasy, the choosing actions from a list thing and having to wait your turn drove me absolute crazy.
I also hate playing in first person, I like to actually SEE the character I play, not just their arms waving in front of me when they fight, do things... First person POV makes at least me feel as though I *am* the weapon the character is playing rather than the character itself.
A good story is also important to me, I want to be able to invest myself emotionally in a game (like I want to with movies, books and shows, as well), so a good story and characters that I feel a closeness to always helps.
I'm also a big fan of options, being able to explore, do sidequests, getting to choose the way you travel, how you fight, etc. Probably why I'm such a big fan of Fable, the game is all about options.
I also love magic, so if there's a magical element to the game, that's a major plus for me.
After that comes things like a decent difficulty (it's equally annoying if a game is too easy as it's too hard - if it's too hard, you risk getting stuck somewhere and never getting past it, if it's too easy then it gets boring easily), decent graphics (it's not the most important thing, but it should at least look decent for whatever style the graphics are in, be it realistic, toony or something inbetween), decent music/sounds, etc.
Games that great replay value, great gameplay elements, and a good story. The graphics could be good, but I'm not one of those people who judge games by graphics. Usually, the games soundtrack is extremely memorable. I also like it when there is a lot of content, like the game is not just to beat the Final Boss, but there are sidequest.
I have to have a great storyline above all else to keep me interested otherwise, I'm going to get bored and stop playing. The gameplay has to be good as well or have something that makes it interesting, like upgrades to your character.