Federalism is a type of government in which the power is divided between the national government and other governmental units. It contrasts with a unitary government, in which a central authority holds the power, and a confederation, in which states, for example, are clearly dominant. This is the essence of enumerated vs. implied powers: enumerated powers are those things that the Constitution explicitly says Congress can do (in Article I): levy taxes, regulate commerce with other nations, borrow and coin money, establish post offices, raise an army, and declare war, among other things.
But Article I also says that Congress shall have the power to "make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers." From this "necessary and proper clause" Congress argued that it had implied powers to do those things necessary in order to achieve its enumerated powers. For example, if Congress has the power to coin money, it's implied that Congress has the power to set up mints and pay workers to run those mints. In McCulloch v. Maryland, the Supreme Court confirmed that Congress can exercise these implied powers.
Although this seems pretty straightforward, it gets more difficult to decide just what counts as an implied power if you consider how to define "necessary." What if when your parents said they'd pay for you to get ice cream with your friends, you deemed it necessary to rent a helicopter to fly to the ice cream shop in style? Was that absolutely necessary, or was that just taking advantage of the opportunity? This conflict over the limits of federal power continues today.
