Home rule is control of an issue at the local level, rather than the state or federal level.
Each of the 50 states differs as to the extent of home rule (as opposed to "Dillon's Law", a legal theory which opposes it). For example, Texas allows home rule for cities exceeding 5,000 population, and for school districts with voter approval, but does not allow it for counties or other special districts.
For example, in education it is important whether an area has the advantage of "home rule," or must follow a state or federal standard. States that have the best public schools tend to be home rule states, such that local school districts decide what to teach (although they are expected to follow general state standards).
Home rule makes it more difficult for supporters of Common Core to impose it statewide in those states.