If you've ever been to Chicago, you'll recognize the landmarks of the city, though they've all been tweaked a bit:
So yeah, this is definitely Chicago. (You can also tell because Tris and friends are taking improv classes at Second City and enjoying deep-dish pizza when they're not fighting.) For a few more pictures, check out Veronica Roth's blog, which has a whole category of posts about Chicago.
And yet, for all its Chicago-ness, the city where Tris lives isn't the Chicago we know and love. For one thing, there's a giant fence that locks everyone into the city (11.43); for another thing Lake Michigan has become a swamp (12.57); and for yet another other thing, large parts of the city are crumbling or in ruins. It's now "a patchwork of new, clean buildings and old, crumbling ones" (3.35). We never learn what happened to Chicago, but we can be pretty sure that it wasn't so good.
Of course, a dystopian setting needs more than just some crumbling buildings. It needs some weird new government and society. Since Tris isn't one of the leaders, we don't see firsthand how the government works. But we do hear about it, so we know it's led by an Abnegation council (because selfless people aren't likely to be power-crazed maniacs) with representatives from other factions coming in to speak, but not vote (4.41).
We also hear how society is organized so that each faction takes care of some particular part of the city; so, for instance, the Dauntless are the guards and security of the city (11.56), while Amity seems to be in charge of the farms (11.46), because friendship is an important part of picking fruit.
But the big change between our time and Tris's time is the separation of the city into the five different factions. How long do you think that would take? How far in the future are we talkin' here?
This is the part that gets a little more confusing. Because Divergent is clearly set in some other time than ours, but it's not clear how far in the future, or why things have turned out the way they have. For instance, Tris remember that her mother told her that people used to avoid genetically engineered produce, but now that's all they have (4.26). So, what is that, 20 years in the future? 200?
We really don't get a lot of info about the past or how this city came to be the way it is other than a few hints about the vague apocalypse (or time of troubles) that destroyed parts of Chicago. There are no hoverboards or tablet computers or reprogrammable tattoos, so it may be that the disaster knocked technology down to the level of guns and cars (in other words, our level); or maybe the disaster happened… wait for it… wait for it… now. Is there a disaster outside? If so, we choose—well, can we choose to be Divergent?