Public means funded from the public purse (ie, by government) and controlled by public policies (ie, by government). Non-Public means the opposite. US public schools are either traditional district schools or charter schools. Non-public schools include parochial (attached to and run by a parish), proprietary (owned by person or company and for-profit), corporate (owned by a corporation such as Meritas) and independent.
Why does this matter?
It matters because of who makes the decisions, and thus what those decisions are. These questions will be explored over time, but in essence this means what happens or doesn't; what is funded or not; what is prioritised or not; what is taught or not; when and how something is taught or not; etc, etc, etc.
It matters because different schools suit different students, families and teachers. For you to know if a school suits you, you need to know who is in control and how the school is funded.
This discussion was further developed here.