A charter school differs from a public school in that it does not benefit from property tax dollars, they are operated by a seperate organization that obtained a charter to run a school. The only funding they get from the state is based on enrollment. The rest of their funds comes from fundraising for private donations. They are not allowed it charge tuition. Charter schools still have to adhere to certain state requirements (like standardized testing) but after that, they are free to teach whatever curriculum however they like. They can set their own hours and policies. This leeway allows charter schools a lot a creativity and diversity. The catch is that they run enrollment on a lottery system. It’s your luck if you get in or not lol. One of the top schools in the entire country is a charter school in a Dallas suburb, I’ve been trying for 6 years to get my kids in, no luck 
Magnet schools are specialized schools that work within the public school district. They are for specialized areas/interests like health sciences, fine arts/drama/music, business, technology etc. These are also highly competative, and have rigorous auditions and applications for acceptance.