What were issues that brought attention to the division of power between the federal and state governments during the early decades of the US?

Division of power had been an issue since before the
Constitution was drafted. Indeed the Articles of Confederation were carefully drafted to
protect the rights of the states vis a vis the General government. Obviously, that
government was not workable, and the division of powers came into play again at the
Constitutional convention and shortly thereafter.

The true
issue governing division of power was interpretation of the Constitution. Some, such as
Alexander Hamilton, preferred to construe it broadly and allow the Federal government
any powers not specifically denied to it by the Constitution. Others, such as Thomas
Jefferson and James Madison, argued for a strict construction with power belonging to
the states unless it was expressly granted to the Federal government or denied to the
states. The problem raised its ugly head several times almost immediately after the new
government was formed. Jefferson and Madison argued against formation of the Bank of the
United States; they believed that Congress had not given the power to the Federal
government to form a bank; rather that power was reserved to the States under the 10th
Amendment. It again came to the forefront with the passage of the Alien and Sedition
Acts. Jefferson and Madison secured the Kentucky and Virginia
resolutions which said states had the right to declare Acts of Congress unconstitutional
within their respective borders. John C. Calhoun later raised the same issue in the
Nullification doctrine mentioned in the answer above. Chief Justice John Marshall
attempted to put the matter to rest in the case of McCulloch vs.
Maryland in which he argued that any act of Congress which could legitimately
be tied to the Constitution was constitutional. The Supremacy Clause of the Constitution
states that the Constitution, any Federal Laws or Treaties passed pursuant thereto are
the "Supreme Law of the Land," to the exclusion of any act of the States. Marshall
relied on this clause in his argument; however it did not put it to rest. It is still
around, with conservatives--particularly the Tea Party--arguing for a Strict
interpretation of the Constitution with more power to the states; Liberals argue for a
more broad construction.