Chicago is entirely inside Cook County
Unlike Atlanta (split between Fulton and DeKalb) or several other major US cities, Chicago doesn't straddle a county line. All 84 ZIP codes with Chicago as their primary USPS place name sit inside Cook County, Illinois, and Cook County has been Chicago's home county since the county's creation in 1831 by the Illinois state legislature -- the 54th county established in the state.
Cook County is much bigger than just Chicago
Chicago is Cook County's seat and by far its largest city, but the county contains 135 separate incorporated municipalities in total -- 23 cities, 1 town, and 111 villages, either entirely or partially within its boundaries. Well-known Chicago suburbs like Evanston, Oak Park, Cicero, and Skokie are all their own separate municipalities inside Cook County, distinct from Chicago itself even though they're commonly lumped together as "Chicago" in casual conversation.
Cook County is the second-largest US county by population, and shrinking
Cook County's population of roughly 5.18 million makes it the second-most-populous county in the country, behind only Los Angeles County, California. But Cook is the only county in the national top three that's losing population -- it's down about 81,000 residents since 2020, while third-place Harris County, Texas (Houston's home county) added roughly 106,000 residents in a single recent year and is on track to pass Cook County for the number-two spot.