Using private education as an example, in the United States, each state is empowered by law to regulate educational institutions. Each state has laws that regulate the practices and standards of schools generally, and each state has laws that specifically regulate private schools. The regulatory body -- usually the state's Board of Education -- will administer and enforce state statutes pertaining to registrations, licensure, record keeping, curriculum, special education, safety and student transportation.
Continuing with the example of private education, there are various accrediting bodies that each certify based on a particular set of standards. Two national private school accreditation bodies in the United States are the National Association of Private Schools (NAPS) and the National Private School Accreditation Alliance (NPSAA). NAPS is an association of private Christian schools, so their standards include criteria that are specifically religious, in addition to academic. The NPSAA accredits schools strictly on academic standards.
The line between regulation and accreditation is blended to some degree in private education because states maintain regulations that require certain kinds of accreditation for private schools, not of the schools themselves but of the teachers. Some states have certification requirements that are independent of any accrediting body while others require accreditation from the National Council for Accreditation of Teacher Education (NCATE), a non-governmental accreditation body.
State accreditation bodies, for example, the Georgia Private School Accreditation Council (GAPSAC), sometimes work with regulatory bodies to facilitate aspects of a practice. In this case, the GAPSAC (an accrediting body) works with the state Department of Education (a regulatory body) to ensure coursework in private schools can be transferred to state schools. In this case, the accrediting body applies regulatory standards to the process of accreditation. The GAPSAC is not, however, a state agency but an association of the private schools themselves.