What roles do the various levels of government play in K-12 public schools?
In the American system of government, state, federal, and local levels of government all typically play some role in K-12 education.
Education is an area that is in the jurisdiction of the states and not of the federal government. Our Constitution created a system of federalism in which some policy areas are state concerns and others are federal concerns. Education is a state concern. State governments are typically very involved in K-12 education. The state governments are in charge of such things as the licensing of teachers and the creation of standards that all students are supposed to meet at each grade level.
States can delegate as much or as little power to local governments as they like. Most states give some degree of control of local education to local school boards. (It is important to note that school boards are generally independent of local city governments.) School boards can often decide on things like the curricula for their districts and other details of education, such as how to divide the grades (which grades to have together in a school) and how to schedule both daily classes and the entire school year.
The federal government has taken a role for itself in education through the use of grants and through such laws as No Child Left Behind. The federal government has no official authority over education, but it does give large amounts of grant money to state and local governments. It can impose rules that must be followed in order to receive the grant money. This is how the federal government takes a role in an area that is supposed to be a state matter.