A game doesn't really draw me in unless it has captivating gameplay. Some people might fight beat-em-ups to be boring and repetitive. I find them to be incredibly fun, especially if they incorporate some sort of combo system like Dungeon Fighter Online. I greatly enjoy World of Warcraft as well, though not for its hard-wired gameplay. I enjoy WoW for how I can use the gameplay to create stories and build upon character lore with other people. I play Neverwinter and enjoy it because its control scheme is intuitive and provides a good deal of customization when it comes to character builds. I highly dislike first person shooters on consoles because the controls feels awkward and clunky in my hands, and as such I usually don't play them for long, if at all. Shooters on PC, however, feel much more natural to me and as such I enjoy them better than I enjoy console shooters. Storyline isn't much of a consideration to me, although it is second place in what draws me to a game. I can stomach a bland storyline if the game plays like a charm, but even the most well-written story can't save a clunky game for me.
Graphics aren't a consideration at all. I play NES throwback games and stuff that looks like it belongs on a Commodore 64. And I love every damn second of it because the games control like butter and are incredibly fun to play. I own a computer where it's pretty much impossible to have cutting-edge graphics and a decent gameplay experience at the same time, so I opt for the experience over the looks. I'm sure most people feel the same way, but it doesn't hurt throwing in a few things like that.