Public post-secondary schools are usually operated by a state or local government (or both) and supported by money collected through taxes. Private post-secondary schools are operated as nonprofit organizations. They generally need to receive more money from their students in order to operate, though not all of their students pay full price.
“State” colleges and universities are public schools that offer a relatively low-cost education to residents of the state in which they are located. You’re probably well-acquainted with your state’s big universities. (Roll Tide.) A catch with these schools is that they’re only a lower-cost option for state residents. If you’re from out of state, these schools will charge much more money for you to attend, sometimes as much as the full price of a private college or university.