Public Schools are the normal schools in USA. Schooling till grade 12 are mandatory and free. Schools are primarily funded by the property taxes of the city. Therefore, the schools in affluent neighborhoods tend to be better as they have more money and resources. While people buy their home or decide where to live, quality of the local school district is a big deciding factor. If you live in the school district, you child is automatically qualified to go to the school in the district.
Chartered Schools are also another form of public school. They get public funds therefore they are free to attend as well as they do some other fund-raising activities. Chartered school serve to larger area and in best chartered school, there is some sort of entrance exam and or some sort of lottery. Most chartered schools are focused academically, and they might not have enough resources in Athletics, Band, Choir etc.
Magnet schools are the focused schools in certain area e.g. Math and Science magnet school, Art magnet school etc. Magnet schools are funded publicly therefore free for the students. However, it is usually very difficult to get in. Many highly accomplished kids go to magnet school as steppingstone to go to highly ranked Ivy league colleges. Magnet schools are mainly focused on academics therefore wouldn’t have opportunities in Athletics, Arts, Music etc.
Private schools are privately funded and therefore cost a lot of money. The primary reason why parents send kids to private schools is the religious studies. Most of the private schools are either Christian or Islam. In some parts of countries, the public school is bad which makes parents to send kids to private school if they can afford.
I am big proponent of Public School. My recommendation is to live in a good school district and send your kids to the Public School. It will help your kid to develop overall and have full school experience – academically, socially, athletically. If your kid is good, they have equal chance to go to the best colleges without going to other fancy schools.