I will start by offering two links.
One is to
LostSoul's thread on this forum where he talks in detail about running his modified 4e sandbox. One important feature of LostSoul's approach is that the players have a lot of information upfront about the nature of the world and the challenge level of different areas, and there is no rule about not letting the PCs act in accordance with metagame knowledge. Because the players are able to metagame in this way, they are able to make their own choices about where they go in the world and what quests they take up.
The other link is to
a thread on The Forge - the post by Paul Czege, about halfway down the page, talks about how he frames a scene - he is very aggressive in how he sets it up (which is sort of the opposite of AbdulAlhazred's idea of 2 or 3 hooks that the players may choose from) but the way the scene resolves depends on how the players respond to the stuff he throws at them in the course of the scene.
I especially like the idea that he shapes the NPCs' motivations during the course of play, in order to building meaningful responses to the players' choices. This is different from a sandbox approach, but is an alternative that I think can work well.
I have a number of overarching themes to my world that I keep in mind.
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I use character backgrounds a lot and keep on bringing characters pasts back to haunt them. I also try and tie what they wrote for their backgrounds into what is happening now, or what might be happening in the near future.
In short:
Give lots of options
Use character backgrounds as part of those options.
I agree with this, especially about grounding the PCs in the world.
My own approach is to work with the players in designing their PCs and incorporating their backstories into the gameworld, and then to present situations that will speak to those backstories. So, for example, if one of the PCs comes from a city destroyed by goblins, I will present him with goblins threatening a human village and see what he does. I will have him meet his mother, now an enslaved servant in a goblin fortress, and see what he does.
What I've found works for me is to not punish the players for the choices they make, or to be judgmental about it (which means, for example, be very very careful using alignment, or gods, or law enforcement, in an aggressive way to direct your players' choices). So to continue with the above example, if the PC is fighting the goblins and they surrender, what does he do? Suppose he decides to execute them on the spot - this tells us something about the character, and it changes the PC's place in the gameworld - maybe the villagers he's rescuing cheer as he decapitates the goblins, maybe he gets a reputation for being bloodthirsty, maybe his mother can't believe what her son has become. Any of the players' choices should lead to repurcussions that build future encounters and future adventures. They shouldn't feel that they have to make the "right" choice (ie the GM-sanctioned one) in order to be able to keep playing the game with the PCs they want to play.