This question is about how government shapes society's school policy and practice. Public education in the American system is financed and run by the federal system and state systems. Funding for schools comes from state property taxes, with federal funds for special programs.
The question is about branches of government, and is not an easy one to answer in short space because the educational system is run on multiple levels and is highly complex.
State government affects policy and practice primarily from legislative influence, as laws proposed and passed by the legislature affect school funding, but the executive can have a hand in withholding or providing funding. The judicial branch of state government has a much smaller role in school policy on the state level.
On the federal level, all three branches are involved in policy. A good example of this is the 2001 No Child Left Behind law, which was pushed by the executive (Bush administration) to reform schools. Ultimately, No Child Left Behind had unintended consequences and many of its provisions were disbanded in 2015. Arguably, the legislative branch, which had bipartisan support in Congress and Senate, had the most involvement in passing NCLB, and the most involvement in disbanding it, although local and state activists and elected officials played a larger role in reform.
As for longstanding educational policy and practice, the federal judicial system has played a large and, debatably central, role in education. The reason for this are programs like school lunches, civil rights reform, and special education funding. Many of these policies developed from civil rights legislation, sweeping reform such as the War on Poverty (an executive and legislative program) that involved judicial review.
The most straightforward answer to this question is, the legislative branch of government has the most extensive effect on school policies simply because they pass the most laws and control funding. Regulations are controlled by the executive branch through the Department of Education, and defining the laws occurs through this branch, as well as the judicial branch.