In the United States, a public school is a government-funded school that accepts all children in an area. This stands in contrast to private schools, which are privately funded and have fewer restrictions on whom they accept and what they do.
Since public schools are run by the government and paid for by taxes, they are bound by US laws that apply to the government. The most important of these is the First amendment to the Constitution, which prohibits schools from favoring one religion over another, or religion in general over none.
At the same time, the free exercise clause guarantees freedom of religion. The need to strike a balance between it and the establishment clause means that students are free to worship, but that the school itself must remain neutral. Thus, for instance, students may pray if they wish, but teachers may not lead a class in prayer.