Federalism, at is core, is a system of government in which power is divided between a national government and other governmental units. In the case of the United States, federalism is represented by the federal government and individual state governments.
The complex push-and-pull between these two sources of power has been a debate in the United States since its founding. The Articles of Confederation (1781) were the colonies' first attempt to reconcile the issue of state vs. federal sovereignty. The Articles, however, relied on an honor system between the states that said they would convene meetings in order to make decisions about fundamental issues like taxation, disputed territories, and foreign policy, since control over these decisions was not given to a centralized government. Many politicians, including George Washington, believed that the powers given to the federal Congress in the Articles of Confederation were simply not enough to ensure a functioning nation. Their solution was to convene a Constitutional Convention to come up with a new document to enshrine the American system of federalism.
The debates at the 1787 Convention were wide-ranging and extremely personal: state sovereignty and representation in government were open wounds for a nation that had just liberated itself from a monarchy. The Constitution itself was a major compromise between the Federalists (a political party that wanted a stronger central government) and the Anti-Federalists (a political party that wanted a weaker central government). Both parties wanted a system of federalism, but the balance between states' rights versus federal power would be an ongoing debate—one the Constitution purposefully left ambiguous.
One of the most frequent modern critiques of the Constitution is the Electoral College system. Essentially, in this system, "electors" were voted upon by individual states (in a number proportional to that state's number of representatives in Congress), and then those electors would vote for president. There was logic to this: keeping the executive branch independent of the other branches and separate from the whims of the individual voter was a way of preventing corruption. Controversially, though, as we've seen in both the 2016 and the 2000 presidential elections, it is entirely possible for a politician to receive the most citizens' votes and still fail to secure the majority of Electoral College votes.
The method of appointing senators and congresspeople is another debate that has continued even after the Constitution's completion. During the Convention, the issue divided small states (who wanted equal representation) and large states (who wanted proportional representation).
Ultimately, a compromise was made: the number of representatives a state had in the House would be proportional to a state's population, while each state, regardless of its population, would have two senators.
The result of the compromise on Congressional appointments ultimately undermined the stated goal of the Constitution: to provide equal representation to each state in the federal system. The system of having two senators per state, for example, created a system in which partisan senators could effectively block federal legislation, even when the legislation is popular among citizens. In other words, senators from less-populated states end up having a significant amount of power in making legislative decisions.
One recent example of this is President Barack Obama's attempted gun control legislation in 2010. While its primary tenet—universal background checks—was polled as incredibly popular among American citizens (with a 90% approval rating), the Senate, in which less-populated states (where gun control is typically less popular) have proportionally more power, had the ability to block a measure that was approved by a majority of American citizens.
Of course, the Constitution was never intended to serve as a perfect solution to the question of how power is balanced between state and federal governments. Instead, it was left purposefully ambiguous, open to changes as "We the People" see fit. Determining the proportions of power and the relationships between state and federal governments is an ongoing responsibility of the American people.