Public in what sense? Elementary through Secondary aka K-12 is, but is mandatory for students unless they take extra-ordinary measures to not be there, e.g. homeschooling, or dropping out at 16. Funding through this is usually handled on the state and local level, often by local property taxes.

Public Universities are ones funded by the state, to varying degrees.

Private Unis have their own funding, usually through a mix of tuition and large endowments.