Ranging from back-to-basics to Montessori methods to schools for disabled kids, with a hundred other models in between, charter schools are a hybrid: public schools with some features of private schools. As public institutions, they're open to all who wish to attend, paid for with tax dollars, and accountable to public authorities for their performance (especially student achievement) and decent behavior (e.g. non-discrimination). But they also have features associated primarily with private schools: self-governance, freedom from most regulations, the ability to hire whom they like (usually without a union contract), control of their own (secular) curricula, and attendance only by youngsters whose parents elect them. Besides upward-accountability to state or local authorities, they must satisfy their customers--or they won't have students.
According to Education Week on the Web, charter schools are schools run independently of the traditional public school system but receiving public funding, run by groups such as teachers, parents, or foundations. Charter schools are free of many district regulations and are often tailored to community needs.