Education in the United States is typically broken up into five phases:
The great majority of students in the United States in elementary school and high school attend public schools. Public schools do not charge tuition (they're funded by taxes, so they're not exactly free) and are open to any school-aged children in the school district.
School districts are an important thing to understand when moving here. The United States has some terrific public schools, but it also has some terrible public schools. The school district you live in determines which public school your child can attend. You'll typically find that a home located in a good school district is more expensive than a home that isn't. It's also important to know that school districts don't necessarily overlap with an entire neighborhood – some neighborhoods may be split in half by the boundary of a school district, creating a huge difference in home prices even though the homes are separated by a few feet.
And some school districts may have a great elementary school but a bad high school, or vice versa. So if you have school-aged kids and plan to send them to public school, make sure you check the quality of the schools before you find a place to live.
The New York City Department of Education has a useful website that grades all of its public schools on a number of criteria.
CHARTER SCHOOLS AND PRIVATE SCHOOLS
Charter schools are a type of public school but often have a private sponsor as well to help fund the school's operations. Charter schools typically pop up in communities with bad public schools. They focus on rigorous academics and are less beholden to the rules and restrictions of regular public schools. Because of their strong curriculum and teachers they are highly desirable, which means there are never enough spots for all the kids who want to go to them. Many accept students through a lottery system. This is something you may want to look into if you will be living in a neighborhood that does not have any good public schools in its district.
Private schools are another option for educating your child. New York City has dozens of private schools within its borders; some of the private schools are among the best in the country and send a large proportion of students to the top institutions of higher education in the United States. There is no restriction on which private school your child can attend provided he or she meets the school's criteria (some are very competitive academically). The downside of private schools is that they do charge tuition and can be quite expensive. You're also still required to pay the taxes that fund public schools so you're double-paying in a way. Most private schools do offer financial assistance to students who can't afford the full tuition.