A magnet school may or may not be affiliated with a public school district. It could be public, a charter, or private.
A magnet school typically focuses in on a very specific domain like a School for the Fine Arts, a School for Technology, a School for the Highly Gifted, a School for Science & Mathemtatics, etc. Within the school day at a student will typically receive more hours of instruction in a specialty area, but still receive the other basic instruction like writing, literature, social studies, etc.
Teacher at a magnet school are also specially trained to deal with the curriculum. For example, I currently teach Elementary Gifted students, and am trained and certified in that area. I could easily teach at a magnet school that serves the needs of highly gifted kiddos.
Most magnet school serve a very small population-perhaps less than 200 students depending on the grade levels. Magnet school usually are not sports/athletically oriented. However, students might be able to participate in the athletic program of their home-based public school.