A public school means different things in different countries:
USA and Canada change
In the United States and Canada, public schools are schools that are paid for by the government. They are open to all students who live nearby without any charge. They are usually paid for through property taxes that are applied to everyone who owns buildings in the area around the school.
Public schools greatly expanded in the United States in the 19th century. Every U.S. state now has free public schools.
Some people think that public schools have done a bad job of education in the United States. One solution has been to give "vouchers" to parents that help them send their children to private schools. Critics say this takes money away that could go to improving public schools.
See British public school, where the meaning is different.