Now story, mechanics, and all other facets of a video game (key word here) are important. But before I go further, I must say that the game part of video game is the most fundamental point. There are other forms of interactive media, but what sets apart a video game from all of these is gameplay. Their stories tend to be basic, any games can have great graphics, and lots can settle on gimmicky mechanics (notably the Wii, X-One, and PS4). What sets apart a good game from a bad one can be how much gameplay it offers. This isn't just how long it takes to get through it, because Final Fantasy takes forever to get through but isn't very good gameplay wise.
But then there's another term I need to give my definition on. Gameplay centers around what the game intended to do and how well the mechanics and the game flow together to create that end. For example, Call of Duty has decent (if stale by now) FPS gameplay. It was meant to be an FPS, it is an FPS. It's solid, it's fun. What Call of Duty is not is a stealth game. Nothing is as terrible as a game trying to pretend to be something it isn't. Expectations absolutely have to do with what gameplay offers; if a game says it's a strategy game then you had better hope that it plays as a strategy game. Endless Space is a prime example of what I consider good gameplay. I expected a game about space and turn based strategy, and the game was built around offering that gameplay through mechanics, feeling, and atmosphere.
In conclusion, to me, what makes video games appealing is an expectation on gameplay met and fun to go along with it. The most important thing a video game has to offer is the enjoyment that comes associated with "game." If the world, story, and graphics are all terrible but you, the player, thought the gameplay offered what you were looking for and didn't drag down the fun, then it's good. My over all impression on video games can be summed up best with Han Solo's quote in Star Wars (no suffix, because if you're a true Star Wars fan you'll get it) "Hokey religions and ancient weapons are no match for a good blaster at your side, kid."