Hard to generalize since it will differ from game to game but here goes:
1) Controls - Quite simply put these can make or break a game. They are the most fundamental part of making a game fun to play and can influence everything else such as level design and enemy variables. The unique thing that videogames have over any other medium is the level of control necessary to progress, which is why they should perfect it in order to stand out. If they are not done well they can quickly make the experience frustrating making other aspects that may be well done easily missed.
2) Level design - Likewise with controls, this is something that is unique to a videogame and is what makes a video game special. It is important they are designed well in order to make sure the gameplay can take place in a suitable environment. It is essential that the levels are compatible with the core gameplay and that it is somewhat clear which way the player must go to progress.
3) Visual design - For this segment I'm not including graphics, I'll talk about that later on. By this I mean how the aesthetics were directed, such as enemy, menu, level, animation and character designs. They have the power to create some powerful moments and well directed art can inspire the player to keep playing in order to see more of it. It is also one of the key features that help distinguish games from one another.
4) Sound design - This includes environmental sounds as well as voice acting and music. Sounds can often be used to make gameplay more satisfying and let the player know if what they are doing is effective like sounds made when hitting an enemy. Good and and bad voice acting have their obvious effects, music has the power to add layers of atmosphere as well as making events more emotional and memorable.
5) Story and characters - This is a fairly tricky one to place. Since I mainly play games for the gameplay, having a good story is not a requirement for me to thoroughly enjoy a game. If a story is non existent or barely there then it doesn't bother me because I can focus more on gameplay and other things. However a game choosing to focus on the story a lot can either make the game a lot better, or a lot worse since it would take me out of the gameplay to view a story I don't care about. In short if a story isn't presented I'm not missing a lot, if a story is presented it better be worth my time.
6) Graphics - For me graphics are how good it looks at the time. They don't matter to me much because the appeal of them only lasts for a short while and another game is bound to outdo it at some point. Also I will often say that an old graphically bad game with interesting art direction is more visually appealing than graphically good game with boring art direction.