County programs are funded by the county. (like a city division, but bigger)
County programs tend to attract lower-income patients because they must treat everyone regardless of money/insurance. Private hospitals treat people that pay. County programs tend to be high volume, high acuity with poor ancillary services, but that is an overgeneralization. There are zillions of county programs out there, about half of EM residencies would be my guess.
Many county programs ARE University programs. Being a University program means it is the major teaching hospital for a medical school.