Although the Constitution defines the power of the national and state governments, Constitutional amendments have led to some change in the balance, but so have legislation, judicial interpretation and financial incentives.
Via legislation, the national government forbade various practices used by the states to disenfranchise black people.
Via judicial interpretation, the Supreme Court forced state and local governments to meet demands they were otherwise unwilling or unable to meet (such as forcing reapportionment of legislative districts according to the "one person, one vote" principle.
The national government uses financial incentives to extend its power. Grants give the national government substantial power to induce states to comply with national standards. Categorical grants, targeted for specific purpose, leave recipients with relatively little choice about how to spend the money; block grants, awarded for more general purposes, allow the recipient more discretion.