In a film class I took at Loyola Marymount, the professor once talked about the concept of "auteur theory" -- the idea that the director is the primary storyteller, and that the film reflects the director's individual creative vision. Now, not every director is an auteur, and not every film reflects one person's creative vision, but many of the best films satisfy both these criteria.
The example given in class was this: Finding Nemo earned over $300 million at the U.S. box office. Now, ask the average person (or even the average film student) who directed it, and they're not very likely to know (it's Andrew Stanton). Compare this to a film like Reservoir Dogs. Most film students can readily identify the director as Quentin Tarantino, but Reservoir Dogs only made a few million at the box office. I'm not arguing that Dogs is necessarily a better film than Nemo; rather, Tarantino put more of a signature style into his work, and thus he became closely associated with that style.
There are plenty of other film auteurs out there. Just think about the natural prejudices you might have toward a film before seeing it, if the only thing you know about it is the director. You know to expect lots of explosions and CGI in any Michael Bay film. Stanley Kubrick's films frequently have deliberately-paced dialogue and classical music in the score. Hell, even Uwe Boll has become known for having horrendous acting and lots of gratuitous violence and nudity.
It's equally important for videogames to develop their own auteurs in the same way the film industry has. I've only played a handful of games in which a game designer's "voice" can be heard in the same way a film director's voice could be heard. Probably the most distinct auteur I've become familiar with is Hideo Kojima through his Metal Gear Solid series. (Disclaimer: I've not played Metal Gear Solid 4.) When you compare the games to each other, a clear set of common characteristics can be found.
Each game starts off with basically the same premise: the player takes control of secret agent Snake as he sneaks past guards, infiltrates deep into enemy territory, and eventually faces off with a megalomaniacal enemy hellbent on world destruction. But this isn't what makes Kojima an auteur. Instead, it's the unique way in which Kojima takes this basic premise (which is common to many, many games) and uses it to tell stories that are unmistakably his own.
Taken as a whole, the Metal Gear Solid series is Kojima's critique of the poltics of war. The titular "Metal Gear" of the series is a tank capable of firing a nuclear missile, and it's up to Snake to stop the antagonist from utilizing it. Yet Snake is constantly second-guessing his superiors, allies are crossed and double-crossed, and oftentimes it's hard for Snake (and thus the player) to know if he's really doing the right thing.
To illustrate further, I'd like to take a deeper look at Metal Gear Solid 3: Snake Eater (Spoiler alert!). Set during the Cold War, Snake infiltrates a Russian jungle to stop the mad Colonel Volgin from holding the Soviet Union hostage. Along the way, he encounters The Boss, his mentor, who has defected to Volgin's side. Over the course of the game, we learn that the CIA had planned for The Boss to defect in the first place, so that they could later send in Snake to defeat her and become a national hero. By defecting to the enemy and allowing herself to be killed by Snake, The Boss was in fact fulfilling her own duty to her country. (End spoilers.)
This kind of dramatic reveal is typical to the Metal Gear Solid series, and it's a good way to know that you're playing a game by Kojima. By completely surprising players and going against their expectations, Kojima not only makes memorable gaming experiences but communicates the main theme of the MGS series: that in wartime, you can never really know who your allies are.
Of course, MGS games also have a lot of smaller quirks that make them unique. Kojima always sprinkles his games with Easter eggs, references to old movies (including lots of James Bond references), and meta-humor that involves breaking the fourth wall. Perhaps the most notorious example of this happens late in Metal Gear Solid 2: The Sons of Liberty, where the game gives you non-sequiter messages and tells you to turn the game off. At first, it's easy to take these quirks as Kojima just being weird for the sake of it, but I think they each contribute to the central theme of the game in their own way.
Corvus Elrod asks, "[S]hould video game designers try to remain out of their work, allowing the player to establish their own themes through gameplay?" I think the answer must be taken on a game-by-game basis. In the case of MGS, I think the games are absolutely strengthened by having Kojima weave the player through a story of his own design. But of course it's important to also have games that gives players their own degree of auteurship.
Grand Theft Auto IV is a game that, while containing its own story, lets the player decide how that story gets told. I heard from some players that empathized so much with the main character, Niko Bellic, that they refrained from killing or hurting random pedestrians. They wanted Niko to remain a sympathetic character, so they purposefully refrained from the typical mayhem that GTA is known for. Of course, the option to wreak havoc while also advancing the story is still an option. Metal Gear Solid 3, likewise, gives the player the option of killing as few soldiers as possible.
The games I've played that take the player mostly out of the driver's seat would have to be RPGs. Xenogears is the most egregious example I've come across. There are a plethora of instances in the game where the player can do nothing but tap the X button to continue through dialogue and cut scenes before getting back to the action. Yet, Xenogears has a great story, one of the most epic of any game. My issue with Xenogears is not with the story, but how the game chooses to tell it. When I played it, I wanted to feel more like a player and less like a reader.
The uniqueness of auteurship in games is that they can engender players to be authors themselves. The best stories in games are the ones that allow the player to feel like the agent of change while nonetheless acting out the type of story that the designer intended. It's a hard balance to achieve, but I think games like Metal Gear Solid 3 and Grand Theft Auto IV are the closest that games have come to striking that balance.