What argument against British rule did Thomas Paine make in Common Sense?

Although the colonies were fighting the British in January of 1776, when Paine published Common Sense, many colonists were still uncertain if independence from Britain was a good idea, thinking it might be better to remain a colony, only on more favorable terms. By making a cogent argument in simple, understandable language for why the United States should—and must—be independent, Paine helped galvanize support for the Revolution.
Paine first argued that hereditary monarchies and aristocracies are inherently tyrannous, placing too much of a burden on the rest of the people in a nation. The colonies, he said, had no reason to be part of such a cumbersome governmental system. He rejected John Locke's idea that a parliamentary government provided a sufficient check on monarchial power, asserting that monarchs were parasites who had nothing to do but wage wars that other people had to pay for. Paine's answer to this problem of parasitical tyranny was simple: dispense with the British monarchy.
Paine also raised the issue of why the colonies owed allegiance to Britain at all, given the number of colonists that were not of British descent and the number of people of British descent who had never been to Britain and had no direct ties to that country. He asserted, too, that the colonies had sufficient military resources to defeat the British.
Paine argued that a clean break would allow the colonies to establish a republic, with democratically elected officials, which would be less tyrannous and burdensome to the American people than staying part of Great Britain. He asserted that independence was the only way the Americans would prevent themselves from being endlessly plagued with interference from British rulers and aristocrats.

Common Sense is the best-selling American book of all-time. Thomas Paine used his persuasive writing to convince the colonists in Colonial America to fight for independence from Great Britain. This might not seem like a huge stretch by our modern eye - how hard could it have been for thirteen states to demand separation from a country across the Atlantic Ocean when telephones didn't even exist?
The problem existed not in the difficulty of separating, but rather in the difficulty of convincing the colonists that separating was the correct decision. Imagine what would have to happen in modern day politics for Hawaii or Alaska to separate from the United States! It would be next to impossible because you could not convince the residents of Hawaii or Alaska that fighting against a superpower (America) is worth their independence (because life for them isn't all that bad anyways). Now transition that to colonial times, when some of the colonists wanted to become independent of their superpower (Great Britain), but had to convince the colonists that their not-that-bad situation was actually really bad and worth starting a war over.
In came Thomas Paine. He wrote a pamphlet that was mass-produced and spread across the colonies and read in streets and at bars and in bedrooms by candlelight. Common Sense highlighted the issues that the colonies faced--issues that the colonists themselves might not have been aware of. Paine called out Britain on their tyrannical practices against the colonies, and he criticized the treatment of the colonies by the far-off, less than sympathetic King.
Paine then continued on to discuss the issue with calling someone a "King" because it placed them higher on the hierarchy of man, when in reality, he stated that all men are created equal. He suggested that by having a Parliament report to the King, the King overruled the Parliament at some point, and was thus overly powerful.
Thomas Paine struck the hearts of the colonists by examining the offenses that England made against the colonies, and argued that independence from Great Britain was the only solution to their problems. He outlined the idea for a congress with delegates from the different areas of the colonies and elected representatives chosen by congress to lead as president.
Paine also spent time explaining that the colonies possessed adequate power to fight the Royal Navy of England, which solidified the possibility that the colonies could succeed in succeeding from England. 
The practical tone that Paine wrote with worked to coerce colonists far and wide that becoming independent was not only possible, but necessary. Common Sense was one of the most impactful pieces of writing ever written, and because of its legacy, it is still in print today.