Updated: Nov 17, 2020
When the United States was first formed, the founding fathers wanted nothing to do with the British form of government which was known as “unitary.”
Unitary (uni- meaning "one") meant that all the power came from a centralized national government (UK Parliament) and was rolled out down to the local levels.
So their first attempt was the opposite form of government, a confederation. The idea was that all the power would be at the local level in the individual states and be delegated to a weak central government at the states’ choice.
It was a disaster with lots of fighting between the states, debts they couldn't handle, arguments about what type of money to use, and more.
So, the solution was to find a middle way, a form of government in which the powers were shared and balanced between the states and national interests.
Alexander Hamilton and some of the other founding fathers called for a new Constitutional Convention in Philadelphia in 1787, this is where the Articles of Confederation were ultimately thrown out in favor of an entirely new form of government.
That compromise, woven into the Constitution and the Bill of Rights, became known as federalism.
Under this new Constitution, the state legislatures retain much of their rights to pass laws as they see fit, but the federal government also has the power to intervene when it suits the national interest.
Article I, Section 8, of the Constitution contains a list of all of the “enumerated” powers assigned to the federal government. Things like the power to declare war, maintain armed forces, coin money and establish a Post Office.
In that same Section there is also what is often called the “Elastic Clause” that authorizes Congress to write and pass any laws that are “necessary and proper” to carry out its "enumerated" powers. A nice vague statement which leads to some arguments.
So, these powers are known collectively as “implied powers” and have been used by Congress to create all sorts of things like the national bank, collect a federal income tax and set a federal minimum wage, among others.
While the Constitution doesn’t lay out in detail the state's powers, the founders included a catch-all in the 10th Amendment (added in 1791):
“The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.”