Indeed, as UA says, it varies.
For example, in Maryland and Virginia, school districts are coterminous with the county.
But in Maryland, you can live in a town and a county - like Rockville, MD or just the county. Bethesda, MD, despite being a prosperous and well-known place, isn't really there. It's just a "census designated place." Anyone in Bethesda is served only by the county. A relator could call where I used to live North Bethesda, to make it sound posh, when it was really just in the undesignated sprawl between Bethesda and Rockville. There's a town near where IMP lives (in Chevy Chase - which is a real place, but I'm not sure if he's in the real part of it) called Kensington that for years I thought didn't really exist. Then one day I got lost and found myself at an intersection with a sign saying Welcome to Kensington. It was like Brigadoon.
In Virginia, you're either in a county or in the city, but not both. For example, Williamsburg is a city (so-called. It's not very big). James City is the county next door, but Williamsburg is not in James City County.
In Pennsylvania, you live in a municipality - borough, township, city, or, in the case of Bloomsburg, a town - and also a county (except that Philadelphia is both a county and a city). That is totally separate from the school district, which usually encompass several municipalities and have their own tax authority. Water/sewage and parks are usually also coordinated between several municipalities, but not always the whole county.
The county government, such as it is, does the courts, various social services stuff, and some assorted planning things, but the municipalities or the state do everything else. Some rural municipalities are wise enough to leave functions like policing to the state. But some municipalities have only one or two people in their "Police Department." It's a terribly inefficient and redundant system. Roads don't always line up at municipal borders, urban or suburban areas that ought to coordinate development planning don't, you've got multiple people being paid to do a job that one of them could do more efficiently, etc, etc.
In Massachusetts, the county didn't seem to do much except the courts. It was all about the town/city and the state. The school districts were, as far as I could tell, the same as the town, which had it's pluses and minuses.
Those are the only places I lived long enough to pay attention.
As for Orange - there's an Orange County in New York and Orange County in California and a city in New Jersey called Orange. Are there more?