In the United States there are no hard-and-fast rules, and education is governed mostly by state law and varies from state to state. However, there is a common pattern in places that have a substantial Roman Catholic population.
The "public schools" are operated by local government (cities and towns), and financed mostly by local real estate taxes. These schools are free to residents of the town. The details can get much more complicated than that--there may be subsidies from state and federal government, and there may be fees that residents must pay for various activities. But basically, the public schools are free to residents and paid for by local real estate taxes.
"Parochial" is an adjective meaning "of the parish." The word "parish" most commonly means an administrative division of the Roman Catholic Church. The phrase "parochial school" usually means a school operated by the Roman Catholic church. The costs are paid for mostly by "tuition," a fee which the parents pay to the school. Thus, Catholic parents who send their children to the parochial school have to pay tuition to the school, but also have to pay taxes to support the public schools. As you might expect, this causes a certain amount of political tension.
Many Catholics choose to send their children to the parochial school, but many will opt for the public school.
Usually, education is supervised by state government and parochial schools are required to meet state standards. The general curriculum and courses are similar. However, Catholic schools will include religious teaching as part of their curriculum, whereas in the United States public schools may not teach religion. (They teach about religions in history classes, and may teach the Bible as literature.)