I hear this question a lot during game design. In a lot of circles, it's kind of a gold standard - what do characters do? It's a shorthand way of asking what the point of the game is, I think, or a way to encapsulate the play experience in an easy sentence.
I think it can be a useful question, but I also think that not every game is set up to answer it. Interestingly, the games that I hear it about the most often (World of Darkness games, because those are the games I spend the most time working on) are some of the games that the question isn't terribly useful for.
I started running the World of Darkness with Wraith: The Oblivion, and one of the issues I had was "OK, what does a session of Wraith look like?" It wasn't that I didn't know what characters "did". I got that from the text; characters could do any number of things. They could interact with their Fetters, pursue their Passions, or they could become involved with the politics in Stygia if messing with the Skinlands wasn't their thing (I very rarely dealt with Stygia politics, for the record). Finding things for the characters to "do" wasn't hard, but figuring out how to start, and what might happen within a session wasn't immediately intuitive.
Some of that is because the two RPGs I started with (Marvel Superheroes and Chill) were very genre- and mission-focused. Marvel was a superheroes game, and I was playing it in elementary school and high school, so we were doing basic comic emulation. Lots of fights, lots of grandiose scheming by villains. Transitioning to Chill was kind of strange because when I started, I wasn't as familiar with horror, but Chill had an awesome bibliography/filmography section, so I wasn't hurting for reference.
Plus, both of those games had something that later games really don't - prewritten scenarios out the wazoo. Chill included at least one in every sourcebook, and Marvel had all kinds of "modules" published. Sure, they were usually written with particular heroes in mind, but that was easy enough to fix (one of the reasons I didn't buy more books for Marvel Heroic Roleplaying, despite being a big fan of the game, is that I don't have any interest in running or playing established comic characters), and some of the best ones focused on a sub-genre (street level, cosmic power) and let you use your own characters. Chill, meanwhile, showed good use of timelines (as in, how things progress if the characters don't get involved).
Getting into the World of Darkness was jarring, then, because it gave me a much more intricately imagined world, but asked me to populate it and decide what was important. Where the games I'd been running were plot-focused, now I was being asked to make the games character-focused. I didn't know it, but this was part of a larger evolution happening in RPGs, moving from the "go out and kill things and get XP and gold" mindset of Dungeons & Dragons into the "tell a cool story" mindset of later games.
Bringing this back to the question of what the characters do, I think that question is actually more useful for XP-focused games like D&D or mission-focused games like Chill. Both games are about the events that happen to the characters, rather than the characters themselves (and yes, I'm aware that either game can focus on the characters, but I'm talking about the games as presented in their books).
World of Darkness games, generally, ask for a more character-focused experience, and as such I think "what do the characters do?" isn't as useful a question. I saw someone on a forum recently posit that Changeling: The Lost has trouble with that notion, because the answer to the question is "hide from the Gentry." That's a really reductive analysis of Changeling, but it does kind of highlight what I'm talking about. You can cook any World of Darkness game down to a one-sentence mission statement to answer the "what do they do?" question, but it rarely provides enough of a hook to get players involved, at least in my opinion.
Let's take Vampire, for example. Vampire has, in most of its incarnations, been a difficult game for players to wrap their brains around, because vampires are monsters. So, "what do vampires do?" Vampires feed on people, and they play politics.
Now, I've run a lot of Vampire over the years, and those statements don't even come close to covering what I've seen players do in the game. They feed, sure, but that's an atmosphere-establishing scene at best. Playing hunting and feeding scenes was fun when we first started playing Vampire (back in 1997 or so; remember I started with Wraith), but after a while you just start abstracting feeding scenes because they just take a bunch of time if you play through them. That's interesting, though; it's an answer to "what do characters do?", but the novelty wears off.
How about "play politics?" First of all, that doesn't tell you much on its face. It's not like "play politics" is an activity with much definition. Vampire: The Masquerade had a particular feel politically because the characters were all Camarilla (I mean, Sabbat games could be political, but the politics tended to be different and involve more beating people with shovels). Requiem, on the other hand, brings in different covenants and asks the group to figure out what's true in a given area. In both instances, though, "play politics" is a very general thing, and the specifics, the answer to "what do the characters do?" need to be determined in play.
I think that's actually the crux of it. The relevant question for a lot of games, World of Darkness in particular, isn't "what do the characters do?" but "what do these characters do?" That might seem like a cop-out, but I think it focuses the attention on the parts of the game that need it. It tells you that it's not enough to make characters and drop them into a setting and let them go. You need to create enough of the setting that they have something to interact with.
This is why I like games that include collaborative setting creation. Dresden Files RPG has a really good city creation system, but most Fate games actually fall into this category, as do *World games and a lot of the indie stuff I enjoy. It's not just because it takes some heat off the GM, although I definitely do appreciate that, too. It's because if the players help create the setting, they know it. They're already hooked in. They know what there is to know. The alternative is to be told, either in a big infodump at the beginning of a game (which the players promptly forget, in my experience) or to be told during play (which leads to awkward moments where characters don't know things they should know, because the players haven't been told).
I've always been of the opinion that RPGs are best when the players become involved, when they make decisions about the setting and the world, and when they take their characters and apply them to the setting, rather than having the setting happen around their characters. I think if you look at the design work I've done (curse the darkness, Promethean: The Created, Demon: The Descent, and now Beast: The Primordial), it's always been a priority for me, even if I haven't articulated it as such. Get the players involved. Make the game about these characters, rather than any given group of characters. That means giving more weight to the setting and the themes of the game. Maybe the question is better phrased as "what is there for the characters to do?"