After a few decades of playing video games on a regular basis, I've come to the conclusion that there are essentially three elements that make a great solo video game. The reason I highlight "solo" is that in multi-player land, there are some additional elements that I think make the game work and that may weigh differently against the elements I present below.
The three elements, in equal weight, but in ORDER of impact are production value, mechanic, balance.
Production Value here refers to the overall look, sound and feel of the game. It is that glossy look that says "I'm a state of the art game and not something made in 1996" (which, really, was not that long ago... but remember that some land mark games came out around then such as Quake 1 (1996) or Myst (1993) or 007: Goldeneye (1997) ) This "production value" however, goes far beyond the graphics engine to also the audio production (my department!) and the games performance. This last bit is something I often see overlooked in that the game may create great still screenshots, but if it doesn't run consistently smoothly in frame rate, audio cues, etc, the player will quickly tire of waiting for the game to "catch up" to them.
I recently played a fantastic time-management casual game called "Ranch Rush 2" on the Mac. I found later than an iPad version had come out and since the game was mostly a click-click-click type of game as you direct your character around your ranch to harvest bananas, sheer llamas and so on, this seemed like a perfect game to port to the iPad. Unfortunately, this led quickly to frustration as I found the rapid tapping required that was simple with the mouse on the PC was hugely frustrating because of how the game would strangely allow you to click "in between" objects, which would allow your character to travel there... but then do absolutely nothing. This quickly diminished the fun of the game as it no longer was a matter of the players skill in "time management" but of "how precisely can you point your fingers on the screen?" Oy.
Animation should also not be overlooked, as should a balance between very capable 2D and 3D artists. I realize that is becoming more and more common for game makers to save costs by purchasing pre-build 2D or 3D "sets" to build their games on. This is perfectly fine, so long as you can be sure to match them very, very well to resemble a game built from scratch. Make sure to get a second set of eyes to look at your finished product to see what they think about the art. Although we are all taught never to judge a book by its cover (or a game by its screenshots) we all do, and so the investment and effort is very worthwhile in this department.
Once the art and animation have been absorbed by a player, the audio component, almost invisibly, will make or break the next impression. Good game audio is similar to good film audio in that is should help immerse the player in the game without drawing unwanted attention to itself. It should be a driver of emotion and not used as a highlight on its own (exception, of course, for so-called "music games" like Rock Band, Guitar Hero, etc.). Horrible voice over, cheesy or low-fi or misplaced music can ruin an other wise promising game before it begins. Even sound effects placement, a job taken on by some indie developers, is not as trivial as it sounds as it often needs to be subtly shaped to match exactly the animation that is present in the game. This is not always a trivial task.
Mechanic. Many game developers associate game mechanic with very simple casual games, but this applies to ALL games - its just that larger and more sophisticated games may have multiple mechanics. Take a recent favorite of mine - the Gears of War series on XBOX 360. At first blush, this game looks like "oh, another first person shooter", but it introduced a few very interesting game mechanics that will surely (if they have not already) be copied by others. One of the coolest is the way that the player can press a button to dash towards cover and essentially crouch in order to reload and so forth and then press another button to come out of the crouch and then fire on the enemy. Then, when done firing, the player goes back into a crouch. Similar types of strategy had been used by FPS players for years, but Gears of War created a way to make this process less cumbersome and more precise thereby introducing a new mechanic to this common move and thereby greatly improving the enjoyment of the game.
If the mechanics you are asking your player to perform are cumbersome or too repetitive, you will hopefully notice quickly. The mechanic has to be easy... but varied. No matter how gorgeous your graphics and music are, if the player has to click in a circle the whole game, things will get boring quickly. So think about what you are asking the player to do.
Balance This element is also sometimes overlooked. Game makers who put together a cool mechanic with terrific production values are there saying - "hey! It looks like a game and feels like a game! Whip up some levels and let's ship this thing!". The problem, is that without rigorous play testing to ensure proper balance, you will NOT present a challenge to your player. Although some folks play games to have something mindless to do, most of us play games as a mental (and sometimes, physical!) challenge. Having us BARELY win or loose by a little our first go at it is much more preferable than making it so easy that you are just "going through the motions". Even more challenging is making the balance such that it PROGRESSIVELY gets more difficult as you work through the game. Nonetheless, this is a valuable addition and once the game is completed, it gives the players a keen sense of accomplishment and they remember why they started playing video games in the first place.