Charter schools are run by some private organization: sometimes it's a for-profit corporation, sometimes a nonprofit organization, sometimes it's simply an organization of like minded teachers and parents. Since charter schools are not overseen by elected officials, the state will write a contract which sets standards for academic achievement and financial accounting. The state will then review the charter school periodically. Charter schools therefore have two checks that ensure high performance: if the state determines that the school is failing to meet educational goals, the school's contract will be revoked, and if parents are dissatisfied with the school, they will enroll their children elsewhere.
Typically, a charter school will be proposed by a group consisting of teachers, parents, and community leaders. The exact process for forming a charter school varies by state. First, the state in which you live must allow charter schools. Some states have specific application processes; some have more free form petitions to the state department of education. Check with your state's department of education to learn more.
Proponents of charter schools claim that the structure not only enhances autonomy from oppressive bureaucracy but also increases accountability. The rules and regulations that shape a public school district, charter proponents argue, can cripple innovation in the schools. The result may be an uninspired and uninspiring educational program that fails to challenge students or meet their true needs.
Because they are monitored carefully, they have little room to do poorly. If they fail to accomplish their goal, they are closed. Moreover, because parents actively choose to send their children to charter schools, the school administrators know that if they fail to provide what they promise, parents and students will go elsewhere.
Opponents of charter schools say that they are merely private schools cloaked in a public-school mantle, allowing like-minded individuals to opt out of the public school system at the expense of those schools. This makes it even harder, they maintain, for public schools to excel. Proponents counter that charter schools create a healthy competition that forces school districts to offer more and better services to students in their traditional schools.
One of the more controversial ideas in the public school arena is whether to privatize public school districts. The main problem with allowing a private firm to take control of a public school district, say opponents, is that the emphasis will be on cutting costs rather than enhancing education. For-profit firms claim that they improve schools by streamlining and cutting unnecessary costs. School privatization has been tried in some districts, but the long-term benefits or drawbacks remain to be seen.
For more information, check out this article on Choosing a School, and this section on Compulsory Education.