1. The U.S. has many private schools.
First of all, like everything else in the U.S., the sheer number of private K-12 schools in the United States is positively overwhelming. There are over 29,000 private schools. See Private Schools: A Brief Portrait for an overview of the private school scene.
Secondly, freedom of choice is at the heart of American private education. Many private schools were founded by groups of parents who want their children to be educated according to certain religious or educational principles. Christian, Jewish, Islamic, Seventh Day Adventist, Presbyterian, Baptist, Episcopalian, Roman Catholic, and Quaker schools are examples of some of the denominations which sponsor schools. Disciples of Maria Montessori and Rudolph Steiner have gone forth and multiplied. Montessori and Waldorf schools can be found in most communities.
Private schools further subdivide along residential or boarding and non-residential or day school lines. You will find boarding schools for grades 9-12 and junior boarding schools for grades 1-9.
primary/elementary grades or PK-6
middle/junior grades or grades 6-9
high school or grades 9-12 or 13
Grade 13 is what is known as a PG or post graduate year. Many schools offer PK (Pre-kindergarten) through grade 8 and then pass their graduates on to a high school which handles grades 9-12. These high schools in the private school jargon are known as prep schools or college preparatory schools.
Within these various categories you will find a wide range of educational philosophies and teaching styles. That's why it is very important for you to actually visit the school if at all possible. Make sure that the school's way of doing things and what it teaches meshes with your requirements.
Another possibility is to enroll your children at a school which offers the International Baccalaureate program. The IB is accepted universally and is held in very high regard. If you have young children, you will have no trouble finding a good school as the options are wide and varied with uniformly high standards. Montessori schools can be found just about everywhere in the U.S. although you need to make certain that the school you are checking out is actually a genuine Montessori school. Waldorf and Reggio Emilia schools round out the early education offerings available.
5. Seeking expert advice.
Remember: finding the right school means a happy, successful child. Truly, nothing is more important than your child's happiness.