There are many many different types of hospitals....
A: Government.
County (local government), Public (City, State, County);
B: Privately owned
Investor-owned (profit is distributed among shareholders), Not-For-Profit (owned privately, not taxed, run by a board of trustees, any profits are redirected into the hospital), Private (run by a corporation for corporate employees)
C: Federal Government
Military, Veteran's
Generally, a University hospital is A, a hospital affiliated with a church is B (such as the hospital I was born in, Riverside Methodist Hospital in Columbus, Ohio)
edit: regardless of a hospital being publicly-funded, they still bill you (or your insurance), though quite a bit less than a for-profit hospital.