I've also heard of Montessori, magnet, parochial, and home-schooling. What are the different types, and how are they different?
Public schools are run by a local school district. Typically divided into elementary, middle and high school. Public schools must serve all children who live in the jurisdiction, including children with disabilities. They are funded by tax funds.
Charter schools are a special example of public schools, but they are run by non-governmental agencies. Allowed by state law, these schools get the same tax funds of public schools, but have less government oversight. They were created in response to the "parental choice" movement a few years ago.
Private schools are run by a non-governmental organization. They can establish their own admissions requirements, and, while they can't discriminate, they can take whomever they want. They may also not provide the entire range of K-12 education.
A parochial school is another name for a private school operated by a religious group, and almost always refers to a school run by the Catholic Church. It is a private school.
Magnet schools are public schools that have a special emphasis program (such as the arts, or math and science) with the intention of attracting students of all backgrounds and ethnicity to the school (i.e. a "magnet" for talented youth). They were often created as a way to get white kids to go to predominately minority schools as an alternative to busing.
Montessori is a particular educational philosophy and method. Most often found in private schools, although some public schools (especially elementary level) have adopted this method.
Homeschooling started out as an alternative to public schools for those families who object to the philosophy or reality of public schools. Many were established by families with strong religious beliefs who wanted to control the environment in which their children were exposed to education and belief systems. This started out as a solitary adventure with each family educating their own children. Modern homeschooling includes lots of shared activities and teaching between families and groups of families.