All public schools, including charters, are governed by the state and must adhere to their individual state's standards. Private schools are not required to adhere to state standards and are free to determine their own standards.
Each state is free to create it's own standards, but most have adopted the Common Core Standards for Mathematics and English Language Arts. Some states, such as Texas and Virginia, develop their own standards. Each entity has it's own committee that determines the standards. For the Common Core, the committee is made up of members of the National Governors Association and the Chief Council of State School Officers, and teachers who were involved in much of the process.
Before developing the Common Core Standards, college and career ready standards were developed. This allows committee members to have an end point in mind. Then, the grade-by-grade standards were developed and reviewed. After the feedback had been collected, the standards were revised and released. Then the individual states review the standards.
It is at this point that the states may choose to adopt and ratify the standards. Not all states have adopted all the Common Core Standards, and some states completely reject the Common Core Standards, including Alaska, Texas, Virginia, Indiana, Oklahoma and Nebraska. In addition, states have the option of adopting only some of the standards. For example, Minnesota only adopted the English Language Arts standards.