First there is a major difference between "publicly held" corporations -- those whose shares are traded in on a securities exchange or in the over-the-counter market -- and corporations formed for a public rather than private purpose. Some of the latter types of corporations include:
a public (municipal) corporation (e.g., city or town),
a quasi-public corporation (a private corporation that provides public services, such as a water or an electric company), and
public authorities: corporations formed to perform a government services (e.g., a city parking facility).