I was pretty young when I started playing D&D, and lacking people to play it with, I spent a lot of time making characters. I don't think this was too uncommon a practice, but it still sticks with me a little bit. Part of the fun of it was the mechanical element: rolling up stats, picking equipment and so on, but part of it was the fictional element. I'd come up with names, backstories, stuff like that. Never too much - I think I lacked the attention span for more - but enough to make this ranger different than that ranger.
These distinctions and histories did not matter much when I actually got around to playing. They mattered to ME, certainly, as they influenced how I played the character, but no one else was particularly interested, which was fine, because I was fairly disinterested in their stuff too. The adventure we were playing was the adventure we were playing, and we all might die anyway, so what did I need beyond some insight into playing my character?
Things have changed quite drastically for me since then, and the single biggest change has been that those ideas which matter to the individual player have become ideas that matter to the entire game. It wasn't an overnight change, and it's been such a large one that it's almost been difficult to see. To illustrate, let's use a very simple example: a player comes to the table and notes that he's armed with his father's sword. This is going to elicit one of three responses: disinterest, acknowledgment, incorporation.
Disinterest is the classic response. A nod and a shrug, and sooner or later the game going to have a +1 sword in a treasure pile and if you want to pass it up for your roleplay priorities, you're welcome to, but that's your problem.
Acknowledgment is a step up from disinterest in that the GM (and possibly other players) actually try to at least take some steps to not completely screw you over for your bit of history. The GM might, for example, come up with a way for your sword to gain or reveal new enchantments in lieu of gaining a new treasure. No one goes out of their way to bring it up, but they at least try to be considerate.
Incorporation is when the game takes it a step further and makes the idea matter within the context of your game. Perhaps the GM does it directly, by making the sword significant, or indirectly, by making your Father's history with the sword part of the driving story behind the campaign. This is the step when things really round a corner, primarily because response demands that the GM actually be willing to put in some work to support the player's ideas. There won't be much help for this in has no place in published adventures or settings, so it's all on the GM's head.
Note that none of these have spoken to game mechanics in all but the most tangential of ways. This is because there's some correlation between responses and methods of support, there's no clear oneto one mapping.
Now, disinterest clearly doesn't particularly match with any mechanical support, and that's fine, but Acknowledgment actually has some interesting options. D&D 4e mechanically supports acknowledgment, for example, by making magic item enchantments fungible. If it matters to you that you keep using the same sword, the game will support it. Earthdawn had an even more elegant model that let your magic items grow with you, an idea that has seen use in other places.
This is fine as far as items in specific go, but how about for broader ideas like backgrounds or allies? Still plenty of support in the form of games that allow players a certain amount of authorial control as part of chargen. That sounds like a very abstract idea, but it's actually something you'll find in almost any point-based game, as well as many with advantages and disadvantages. By buying allies and enemies or otherwise paying points for character elements that are important to you, you are gaining acknowledgment from the system. Similarly, fuzzier systems (like FATE) may allow you to use existing mechanics (like aspects) to 'plug in' those ideas that are most important to you.
Mechanical support for incorporation is a trickier matter. The rub is that no matter how well supported the idea might be, even if it's utterly essential to the game design (like the city in the DFRPG or the relationship map in Smallville), nothing can force a GM to acknowledge it if she doesn't want to. Where mechanical support for acknowledgment allows the player to force the GM to acknowledge his thing (so long as the GM follows the rules), when it comes to incorporation, nothing can force the GM to _care_. As such, mechanical support for incorporation is all about making it easier to use the player's ideas than it is to ignore them.
Now, I realize that in stating something like this as a progression might suggest that, since disinterest (and the implicit lack of support) is the least desirable arrangement, the best arrangement is incorporation with full mechanical support. The reality is not that simple. While I'll argue long and hard for incorporation as making for more enjoyable play, the question of how much mechanical support it requires is much fuzzier. You can have powerful, playable incorporation without a lick of mechanical support if the GM is so inclined. Assuming that one demands the other is a bad idea - it focuses on the tools rather than the ends, and it can paint other GMs with exactly the wrong brush.