A lot of great games consist of these, but they are not reliant on them as a whole. These elements make up the game for what it is, and even when one aspect is outright terrible there is nearly always a redeeming factor in the others. A title can solely run on, say, gameplay, whereas it might have very poor graphics or isn't backed up by a story of any sort. And this may revolve around any of these traits.
Gameplay/Controls - I consider these to be similar enough that they are the same concept, because the controls affect how the game is played. Anyhow, I deem them to be the most essential quality by defining casual or hardcore for a game. For example, Yoshi's Story, a casual experience, runs on simple gameplay and linear exploration, whereas a 'hardcore' title like Call of Duty centers on the player's awareness and quickness at the controls. But they are more than just difficulty. They also help determine the level of immersion that a player will go through. Zelda always pulls this off with a perfect balance of additional features (free roaming to name one) which assist the [more-often-than-not] compatible duo of gameplay and controls.
Story - It's very tough to get a good fix on what makes a story in video games. There are too many variances in gaming to begin with. You have those that move on a straight-up plotline (Assassin's Creed), those that advance from one scenario to the other through stages/levels (Mario), and those that have virtually no story at all (Mario Kart). My favorites are those which combine a remarkable story with amazing gameplay, as The Legend of Spyro trilogy did. It had some poignant character development as well, which made the saga all the better. A game isn't necessitated to be story-driven, nevertheless, in order for it to be good. Franchises like Kirby's hold up great without a dedicated plotline guiding it through to the end.
Graphics - The world in a game is a top factor to how it will function and come to look like. I've seen really beautiful realms produced aesthetically, from the colorful vision of Okami to the exquisite realism of the Halo series. It's an understatement so say that visuals are only good for making the game look good, because they're main objective is to set the tone, create an atmosphere which is enjoyable to take part in. Unfortunately that is greatly reversed by a majority of gamers who claim that graphics are everything. But they are not everything for me; and nor are they the most crucial element. As much as they're useful for giving the game it's appearance, I've always found more gusto in how it played.
As an adage of mine goes, I would rather play a game with terrific controls and poor graphics than a game with awesome graphics and horrendous gameplay. Story, at its core, plays less of a role than both of these, but it benefits the game best when it doesn't interfere too much with lengthy cutscenes so that the three of these can work together in unison.
