If I'm talking to another Chicagoan, I'll usually say I live in Edgewater, or sometimes "Foster and Sheridan" or "5200 North". Referring to your location within the city by the nearest major street intersection or by your street number seems rather unique to Chicago.
If I'm talking to a suburbanite, then I'll say I live "in the city." (If they ask which city, then I'll give them a dirty look.)
If I'm talking to somebody from outside the Chicago area, then I simply say I live in Chicago. I hardly ever say Illinois, since the state of Illinois outside of Chicago may as well be a foreign country.
Whenever I've lived in suburban areas, I'll usually refer to the specific suburb if I'm talking to somebody from that area, or just round it off to the nearest city if I'm talking to somebody else. Examples: I'm originally from Cincinnati even though I actually lived just across the river in Fort Thomas, and over this past summer I lived in Boston for a few months even though I was really living in Brookline and working in Cambridge.
-- David
Chicago, IL