Writing for video games is far more than putting some text or cut-scenes here and there. To me, the game is (or should be) the story. What actually happens during the levels should mean something to the player. One action can bring him one step closer to beating a good game or it can bring him one step closer to saving the living, breathing world he's been living in for some time.
Sadly, when we speak of story in games we (game developers) often refer to the annoying cut-scenes that slow down the action between mission one and two. But I'm happy to say that it's not the nature of video games to be like that, it's just the way we've been making them for some time. Maybe it's due to the current structure used in AAA game development businesses that places the writer as a separate entity like the artist, the programmer or the sound designer (none of which should be isolated anyway). The writers, not really knowing what happens during the levels just fill the blanks between them and try to put as much as they can in order to feel like they've been a part of the game development after all. Then, lots of people play those finely marketed games and get to think that's how games are. Eventually even indie developers, in which the writer is often the same person as the designer / programmer / artist, follow the same « level / cut-scene / level » model.
What's wrong with that? Well... let me be harsh by saying that everything is wrong with that! The gameplay is slowed down, the story is often too long (and thus boring) and both of them do not rely much on each other. Both the gameplay and the story are wasted. What do we have to do then? INTEGRATE THEM!
The game has to be the story. If you want to go crazy and write a huge story for your new platformer or first-person shooter, go crazy and do it. Write a backstory for every character and environment, give a place in that world to every creature or machine you might encounter. Make it detailed, complete and thorough, but KEEP IT TO YOURSELF. You DON'T want to EXPLAIN what the characters' are and think, you want to SHOW it. Everything you write about the game world is important to you (the writer or designer) but what is written is not important to the player. If a character is prone to anger, make him react quickly to anything that happens to him, fists clenching, eyes firing up, etc. If the girl from the park is falling in love with the player, maybe she will look at him intently while he passes by or she will actually follow him whenever she can. If the player is hot on the heels of the bad guy, the bad guy might get nervous and try to throw obstacles in front of the player. This is all very telling about the characters, but it will be even more interesting to the player if this all happens while he plays the game.
There is several ways of making the story part of the game and how it has to be done is really specific to each and every game. However, I would risk stating something radical and say that if you have to create more than one interface in which you push the story, there's probably something wrong.