What is it that makes a character seem important, what warms you to them, in a book it can be a deep back story that intertwines with the character at a present allowing one to understand their actions and feelings. In film it maybe the lighting shot and music that subconsciously makes you feel at the right time. With TV they can stretch it over a series giving you time to understand and warm to the characters on a personal level that evolves with time.
Now with games, they are becoming more like blockbuster epics than just mindless score stacking, so how do they do it? I believe a combination of above with a little more. In a game you can warm the same as a book and view it the same as a film but you are the character, you control their actions. This is where games should out shine the others, action and consequence. Games are becoming more choice based and are now beginning to shape a personal story to the player, alternate endings, the rise and fall of friends all of this is now being implemented to try and create a richer story. Speaking of story, of course this is the most important feature but the application of the other story tools really boosts the experience. I really like the idea of a fully immersive character and plot which can be subconsciously or consciously be manipulated by the player for a unique experience they can share with fellow gamers, ideas like Mass Effects background carry over, which uses the previous games save file to manipulate the plot of the new game I find really interesting. Games like this are what will carry the games into the future, of course we can keep all the mindless violence and gore too.