The federal government definitely plays a role in American education today, particularly in the attached to the funds it distributes to states and to local school districts, the specific rules regulating school attendance are the province of each individual state.
States generally require students to attend public school between the ages of seven and sixteen, unless they fall into one of several exempt categories. The upper and lower ages vary somewhat from state to state and are often tinkered with by legislatures, although no state requires school attend beyond the age of eighteen.
The exempt categories typically include private school attendance, private tutoring by a credentialed teacher and sometimes physical or mental disabilities that make attendance impossible. Successful completion of high school usually also exempts students school attendance, although some states require children younger than fourteen or fifteen to continue some sort of school attendance or regular instruction.
State education statutes also regulate details of public school operations such as the number of days of instruction, the qualifications of teachers, and the course of study required to be provided. Some states apply some of these regulations to private school programs as well; other states leave private schools relatively free from state oversight.
Although homeschooling is legal in one form or another in every state, not all states have statutes specifically covering situations where parents directly provide their children's education. Some states consider homeschooling to be a form of private education, subject to the same rules and regulations governing conventional private schools. Some states regulate homeschooling directly, requiring anything from a simple registration form to lengthy descriptions of the curriculum, complete with lesson plans and texts used. Some
states just want to know when families are homeschooling; others want families to get permission to homeschool from local school officials. Some states require periodic evaluation of homeschooled students by achievement tests or other means, just for statistical purposes or to determine whether students should be allowed to continue homeschooling.
Many states' laws are quite complex, with several options to choose from; often living with a particular law is much different from what a common sense reading of the applicable statutes would lead you to expect, and how laws are enforced can depend on who is doing the enforcing in any given year.
Rather than risking the homeschooling laws of your state, it is best to find out and be clear regarding the home schooling requirements of you state.