You are in a sunlit meadow, surrounded by pastoral green hills in an expansive vacant valley.
You are reading text on a computer screen. You are now aware of the fact that you are aware of the fact that you are reading text on a computer screen.
You are suddenly aware of the position of your tongue within your mouth. You are now aware of your own breathing.
Text, as a means of conveying information, has a powerful ability to alter the reader's cognition to the extent that they (willingly or unwillingly) place themselves within a scene, and even self-identify as a character within that scene . What I'm referring to is the concept of immersion. Text on its own is obviously sufficient to provoke a sense of immersion (see, for example, every piece of immersive literature ever produced). However, in a multi-modal medium such as video games, combining visual input (including text) with a rich soundscape and (possibly the most important ingredient) the ability to interact with the game's story, characters and environments, the potential for immersion is much greater. This also creates a higher incentive for video game developers to do as much as possible to facilitate immersion in their games, a responsibility which is only rising higher as technology and the standards of users increase.
It's like I'm actually there, man...
Now, there are some special considerations when discussing immersion in video games as opposed to other media, and I'd like to make them explicit before going any further. The first is immersion necessity, and the second is immersion type.
To pre-emptively counter the position that video game developers don't need to prioritize immersion because not all games require it, I'll admit right off the bat that immersion is not a quality inherent to every game. Nobody has ever felt a sense of immersion playing tetris, and if you have, seek professional help immediately.
"That's me in the corner..."
Puzzle games as a whole, in fact, often lack any substrate for immersion. However, just because it's possible to create a game lacking an obligation for immersion doesn't mean that a game won't benefit from one. In support of this, look at the general trend in puzzle games since their inception; often story, character and environment elements are introduced that are completely extraneous to the puzzle mechanics, and which exist only to foster immersion and an emotional connection to whatever story element is ostensibly tied to the puzzle mechanics. Look at the incredibly successful Puzzle Quest series, which is basically just Bejewledover and over with aliens and spells.
Hmmm, needs more side quests
The moral is that more (or better) story means greater immersion. It's also true that the more open-ended the story, the deeper the immersion, with more opportunity to experience the game however the player would like.
Putting aside non-immersive puzzle games, there are two types of immersion that exist in video games, and I would argue that one of them in fact can only exist in video games.
With 'impersonal immersion', you're identifying with a particular existing character in a game (or novel, TV show, et cetera), a character with a cohesive personality, context and appearance differing from that of the observer. You're Mario, or Kratos, or whoever. You're observing their story-related behaviour in cutscenes, and possibly you're reacting to events as they do, but the appearance and behaviour of the characters is more or less hard-wired and you're there for the vicariously immersive ride.
Unless you really believe this is you
With 'personal immersion', you're literally putting yourself into the game; i.e. your thoughts, appearance (however you'd like to project it as in-game), emotional and behavioural responses to story events, etc, all within a fictional context. This can apply to avatar characters, by the way, in which the player-character doesn't resemble the player themselves but rather a character the player chooses to identify with, and also makes up most of the appeal of the entire Sims franchise. The important feature is that the character responds exactly how you feel like responding to any situation, even though the character may not look like you (your avatar could be an orc) or be in a recognizable context (your avatar could live in Azeroth). Instead of merely sympathizing with or understanding a character's behaviour, you determine it. This requires open-ended story telling and dialogue options, and ideally a multiplayer (or MMO) role-playing environment; personal immersion is actually a requirement for (the vastly popular) roleplaying servers in MMOs. Giving the player the opportunity to choose reactions to events makes the character's reactions and behaviour more in line with the player's themselves, and the more details the player feels like they're putting into the game, as opposed to the game putting into the character, the greater the sense of personal immersion. Personal immersion is possibly the most 'complete' type of immersion, and one that can only exist in video games due to the requirement of player agency in fully dictating story events.
Games like Skyrim and Fallout 3 attempt to do this with appearance customization, non-linear storylines and multiple (albeit limited) dialogue choices. for another example, see World of Warcraft, a game so addictive its users have collectively wracked up 5.93 million years playing it... and that stat is from early 2010.
MILLIONS of years of this
So you can see why game developers care as much or more about immersion than players: it makes Bejewled into a franchise, and it makes players invest the temporal equivalent of the duration of our species' existence 59 times over on a game from 2004.
Finally, consider the trend in recent years to include morality-based story choices in games. Even when early games presented story options, they were more related to which scene the player wanted to experience next, and had nothing at all to do with the player's own values. But look back to games like Fable and Infamous, and remember that one of their main selling points was the chance for the player to not only shape the story by their actions, but shape the story in the direction of (or even antithetical to) their own beliefs and morals.
Even if it means being a huge asshole
So what does science tell us about immersion? And how can developers use this to create more immersive games?
A recent study has shown that people who empathized with a character found themselves internalizing qualities of that character. This tells us two important things: people can identify with characters sufficiently strongly that an extremely deep immersion experience is possible, and (to whatever extent this is intuitively understood) people will seek out characters with qualities they admire. It also means that the reverse is possible; players might be driven to implant their own prized qualities onto the tabula rasa of an open-ended character.
Also, kids (and probably adults) experience more immersion when the characters are more like themselves. 10-12 year-olds report more video game immersion when the characters were of their ethnicity. I'm not advocating that game characters should unilaterally be cast as some Idiocracy-esque uniracial archetype, but if aren't we unnecessarily screwing a lot of people out of maximal immersion? And if only 15% of video game characters are female, might we expect less immersion (and less appeal) for female gamers? But it's not like there's a disproportionately low percentage of games being bought by women or anything. Oh, wait.
This might also have something to do with it
Take-home message: Make characters with personality qualities that players want to identify with (or allow players to demonstrate their desired qualities through a character's actions), and make these characters at least as diverse as the players themselves, and you've got a shot at maximizing immersion.