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There is no single criterion that makes a country sovereign, but sovereignty in the modern world usually refers to a government's ability to enforce laws over territory. The concept of sovereignty has been debated over the ages and no well-accepted definition has since emerged. There is a difference between legal and actual sovereignty, but governments that are able to effectively enforce laws typically claim to be sovereign and are recognized by foreign countries as sovereign.
The concept of sovereignty is commonly associated with Thomas Hobbes, a 15th century English philosopher. In his 1651 book Leviathan, Hobbes advocates a form of exclusive, absolute monarchy to remove humans from the state of nature. Without a sovereign authority to rule over a population, people's lives would be "nasty, brutish, and short." Leviathan has been met with much controversy over the years, but is considered one of the founding works in social contract theory.
During the Age of Enlightenment, reason, rather than heredity, was advocated as the legitimate basis of sovereign authority. Previous centuries were generally characterized by the sovereignty of religious institutions or ruling aristocracies, and this was rejected by Enlightenment thinkers. The French and American revolutions of the late 1700s both sought to establish the sovereign rule of citizens themselves.
There are two different senses of sovereignty, legal and actual. Legal sovereignty refers to the theoretical claim of a governing body to rule over its subjects. These rules are typically codified in a set of laws. Actual sovereignty, on the other hand, is the degree to which a governing body is actually able to control its subjects. If people do not generally follow an authority that claims to be sovereign, little actual sovereignty exists.
Legal vs. actual sovereignty can be illustrated in the case of the People's Republic of China (PRC) and the Republic of China. Around 1990, both governing bodies claimed legal sovereignty over mainland China and the island of Taiwan. In practice, the PRC only exerted real control over mainland China and the Republic of China only controlled Taiwan. These two governing bodies had the same legal sovereignty, but their actual sovereignty differed.
This issue of sovereignty is also important in international relations. Governments wishing to establish diplomatic relationships with other nations must first decide which governing body to recognize as sovereign. In many cases, there may be only one obvious choice. In the case of the PRC and the Republic of China, however, this may not be an easy decision. Recognizing, or refusing to recognize, the sovereignty of a governing body is a common cause of international disputes.
According to Locke, only governments that can protect and promote the natural rights of man are sovereign. No matter if they can enforce laws. Tyrannies and dictatorships are actually not governments of sovereign nations. Peoples, citizens and even other nations are free to depose these leaders and establish true sovereignty.
You might look to Westphalia, John Locke, and also the principles behind "The Responsibility to Protect."
While you are on track with Hobbes, Rousseau and Locke did the best work on social contract. To that end, governments that don't promote and protect the natural rights of men do not govern sovereignty.
Hobbes noted that men are more apt to suffer than to stir but as noted in the Declaration of Independence "long train[s]" of abuses will lead to the eventual removal of the false sovereign.