The United Nations school gives priority to families affiliated to the UN. Children turning 5 years old before August 31st can enter Kindergarten and if you are eligible, receive 75% education grant from the UN. UNIS has students from Kindergarten all the way to High school. About half of the students are affiliated with the UN.
UNIS is a great benefit for UN staff because otherwise admission to a New York private/charter school is a long process. Besides, private schools test students (even 4 year old) and have all sort of requirements.
Many private schools are girls or boys only. They may cost 50% more than UNIS. Small parochial schools (such as catholic schools) are very inexpensive (1/4 of UNIS) and can be a good option for kids between 2 and 5.
UN Education grant covers 75% of the educational expenses to a limit from kindergarten up. It might also cover 75% of college tuition.
On the website of the NY Board of Education you will find what streets belong to what schools (school zoning) http://maps.nycboe.net/
Just click on Elementary schools (1-5 grade) or Middle schools (5-8 grade) to see what are the zones for each school.