-Genre. If it's a platformer or a fighting game then there are more chances I'll play it than say, an FPS or RPG.
-Graphics. And I'm talking graphics generally, not just technically. So the artstyle and visuals are important to me, and if I dislike them I probably won't play it. I think an interesting game for this to mention would be Little King's Story. I love the childish cartoon artstyle in the Wii game, but the anime moe BS the vita remake has is something I awfully despise and made me stay away from it. Games like SMT: Nocturne and Okami, while both PS2 games, were amazing to look at, in contrast to say Skyrim and Call of Duty despite being technically superior, didn't have an appealing art style. It's also a big reason why I think Halo >>>>>>>>>>>>> CoD.
-Gameplay. If it looks and does play amazingly with great polish, then I will play it. Despite both being platformers and having great artstyles, I played Rayman Origins because of the many options you have playing the game over Super Meat Boy because it was more trial and error. (Not to say SMB's gameplay isn't polished, just not a fan of trial and error.)
Of course there are many ways a game can be appealing to me. I bought Punch-Out and Mario Super sluggers on the Wii because the former is about boxing and the latter is about baseball, my two favorite sports. (I also bought MLB 2K12 cuz of Baseball but it's awful. Getting MLB the show on Vita instead.) However, graphics are a big factor. Here are some games I played based on artstyle alone: (Alright, not really just on artstyle.)
-Rhythm Heaven (absolutely love this one.)
-Yoshi's Island (didn't like it as much as I hoped, however.)
-Wario Land: Shake It!
-Trauma Center
-Little King's Story
-Zack and Wiki
-Ultimate Spider-Man (it was a Spider-Man game with an actual comic-book artstyle.)
-Donkey Kong Country Returns
-Legend of Zelda: Skyward Sword
-Rayman Origins
Storyline and originality. Those are the two main reasons why you might see me play Metal gear Solid games instead of Call of Duty ones, which I despise.
I'm more into storyline, adventure, and action. If a game has a storyline alongside some elements, I'll end up playing it. Syphon Filter, Metal Gear Solid, and Skyrim are some examples of games I like that have those three elements that appeal to me.
I love MGS so much, but come on now. :P MGS storyline is basically just...nanomachines.