What is federalism? What is the continuous discussion/argument over state power verses federal power? Is there a point of apparent contradiction written within the Constitution itself? What parts of the Constitution seemingly direct us in competing directions? Which government should have more power?

Federalism is a form of government that unites entities, such as states, cities, and counties, under one federal government. At the same time, each state and local government maintains its own separate power. In other words, the U.S. Constitution establishes a federal government in which power is shared by the federal, state, and local governments. Article 1, Section 8 of the U.S. Constitution establishes the powers of the federal government by laying out the legal rights of Congress. Article VI, clause 2, called the Supremacy Clause establishes the "supremacy of the federal government," while also granting limited rights to the states.While the early founding fathers, such as James Madison and Thomas Jefferson, supported federalism, the early politicians debated the role of the federal government. Some, such as Judge Joseph Jones, believed in drawing clear distinctions between state and federal jurisdictions. Other much stronger anti-federalists believed that national centralized agencies should not be created at all in order to avoid any conflicts concerning "which level of government was more suited to handle a particular task" ("Federalism," American History: Oxford Research Encyclopedias). Federalists and anti-federalists especially clashed over questions of trade and infrastructure. For example, the South opposed Congress's enactment of the Tariff of 1828 since the tariff harmed Southern trade; therefore, Vice President John C. Calhoun asserted that the Constitution granted states the right to interpret and nullify federal law in order to re-establish balance between the federal and state governments. His assertion led to South Carolina establishing a convention when the Tariff of 1832 was later enacted; "the convention passed an ordinance of nullification stating that the tariffs were 'unauthorized by the Constitution' and where therefore 'null, void, and no law, nor binding upon this State, its officers or citizens'" ("Federalism"). Similarly, federalists and anti-federalists bickered over which government--state or federal--had the right to pay for the development of infrastructure such as building roads and canals.
http://law2.umkc.edu/faculty/projects/ftrials/conlaw/statesrights.html

https://usa.usembassy.de/etexts/gov/federal.htm

The U.S. Constitution creates a federal government of limited powers. The 10th Amendment states: "The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people." This makes it clear that the U.S. federal government is the recipient of constitutionally delegated powers. All powers not delegated to the federal government by the constitution remain vested in the individual sovereign states, and in the people of those states. The constitution creates three co-equal branches of government, legislative (creates the law), executive (enforces the law), and judicial (interprets the law). These branches are mirrored in each state constitution. Through judicial interpretation, the federal government's powers under the constitution have arguably been expanded over the years, thereby diminishing or lessening state power. In Marbury v. Madison 5 U.S. 137 (1803), the U.S. Supreme Court gave itself the power to declare laws (federal or state) unconstitutional. To be clear, nothing in the constitution gives the Court this specific power. McCullough v. Maryland 17 U.S. 316 (1819) gave the federal government the additional power to charter a bank. In South Dakota v. Dole (1987), the Court upheld the power of the federal government to attach conditions to highway funds given to the states. Specifically, the federal government was able to tell the states that if they wanted the funds, they had to raise the drinking age to 21. This allowed the federal government to directly influence state laws beyond the scope of its constitutional powers. In Raich v. Gonzalez (2005), federal law (the Controlled Substances Act) prevailed over state law (California medical marijuana), citing the federal power to regulate interstate commerce. However, the commerce clause did not prevail in United States v. Lopez (1995), regarding the Gun Free Schools Act, or United States v. Morrison (2000) (the Violence Against Women Act) as both were ruled by the Court to infringe on powers reserved to the states. Similarly, City of Boerne v. Flores, 521 U.S. 507 (1997) struck down the provisions of the Religious Freedom Restoration Act that applied to the states, stating that the Supreme Court holds the sole power to define the substantive rights guaranteed by the Fourteenth Amendment, a definition to which Congress may not add and from which it may not subtract. Our federalism system of checks and balances is imperfect, and arguably gives the judicial branch the ultimate lawmaking power since it is the Court's job to say what the law means, including the constitution which is the supreme law of the land. In theory, federal law should only trump state law in matters that are explicitly part of the federal government's powers under the constitution. In practice, it is the Supreme Court that makes this determination, nine justices, appointed for life. And how do they get that powerful position? The President, head of the executive branch, nominates someone to fill a vacancy on the court, and it is then the Senate's (as part of the legislative branch) duty to then give "advice and consent" usually in the form of Senate confirmation hearings, before the President can then appoint the nominee to the court. In this way, all the branches of government play a role in determining the scope of the federal government's power.