There are quite a few ways to legally organize your school from being a sole proprietorship up to full blown corporation.
Most schools seem to be set up as a Limited Liability Corporation (LLC) to help protect their personal assets from any risks related to running the business. But perhaps you are thinking about something different like establishing a non-profit flight school. It’s a good idea to discuss your options with a local attorney and/or tax professional. You can also see a brief description of some other options on the Small Business Administration website here: https://www.sba.gov/business-guide/launch-your-business/choose-business-structure
As mentioned above, most schools are set up as a for-profit entity which would make them a Commercial Operation on an airport. However, we do occasionally talk to people who want to start a non-profit operation to provide flight training for a local community effort (e.g., training underprivileged high school students, etc). A non-profit flight school operates very much like a for-profit school, but can reap the tax benefits of non-profit organization while supporting a worthwhile cause.