A Community bank in the US is “local” in that it takes deposits from people and businesses in your community, and lends to people and businesses in your community.
Specifically these banks are not affiliated with a large, multistate financial holding company. They tend to have most or all of their operations in one state.
They are sometimes called “relationship” bankers as opposed to “transactional” bankers.
While not all are private corporations, most are not publicly traded.