The Founding Fathers often had disagreements over "interpretation" and "intent" of the Constitution—why were the following actions constitutional issues, and who or what ideas were in conflict in each: the Louisiana Purchase, the First Bank of the United States, the Alien and Sedition Acts, and the Virginia and Kentucky Resolutions?

The early Republic was marked by disagreements over the interpretation of the Constitution. For example, the purchase of the Louisiana Territory in 1803 was controversial. Jefferson bought the territory from Napoleonic France, but it wasn't strictly clear that the Constitution afforded the executive branch the power to buy land from foreign entitles. However, some of Jefferson's advisors argued that this power was implied under the power of the executive branch to make treaties with the agreement of two-thirds of senators.
The Bank of the United States was also a subject of ongoing controversy. Many opponents of Hamilton's bank argued that the federal government did not have the power to create such a bank. However, in the case McCullough v. Maryland of 1819, the Supreme Court ruled that the government could create the bank under the "elastic clause" of the Constitution, which reads as follows:

The Congress shall have Power ... To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof.

This clause, which is also called the necessary and proper clause, gives the Congress the power to carry out duties related to their duties enumerated, or given, in the Constitution. The power to operate a bank was an extension of the Congressional power to tax people.
The Virginia and Kentucky Resolutions, written by Jefferson and Madison, declared the Alien and Sedition Acts passed under Adams as unconstitutional. They argued that these laws, which made it more difficult to become a citizen and that allowed the executive branch to deport people who weren't citizens, were unconstitutional. These resolutions put forth the idea of nullification, that states can declare federal rules void, setting up the supremacy of state governments over state governments.