Federalism, in its most basic sense, is a form of government in which there is a balance of power between the local governments and the overarching national government. In the United States this manifested itself as the Articles of Confederation, which focused on the limiting of power in the Continental Congress. This form of government is the basis for the U.S. Constitution and all laws created thereafter. The initial stages of federalism lasted into the 1790s, when the first Federalist Party was founded by Alexander Hamilton.
Federalism, although well thought out at the time, did not anticipate several issues where individual states and federal government shared jurisdiction. U.S. Supreme Court Justice John Marshall took an active role in expanding the role of the national government in the early 19th century with decisions in cases like McCulloch v. Maryland and Gibbons v. Ogden. Roger B. Taney, Justice Marshall's successor on the bench, felt differently and his decisions pushed the federal government back to only the powers enumerated in the Constitution. This "dual" federalism lasted through 1933.
The onset of the Great Depression spelled the end of duel federalism. The "New Deal" implemented by President Franklin Roosevelt expanded federal government further into the lives of the citizens than it ever had been before. This new cooperative federalism raised local governments to the same level of importance as state and federal government. Also during this period, the federal government gained more control over funding by introducing a grants system where the federal government controlled which principalities got funding whereas this used to be controlled by the states.
With the election of Ronald Reagan to the office of president in 1981 came the introduction of new federalism. This was seen as a return of states' rights. The grant distribution rights that had been taken away in the New Deal were partially restored when Reagan implemented "block grants," which the states were allowed to distribute at their on discretion.