What gives a country the right to rule/exist?

"Sovereignty" is the word for a country's right to independently rule its own citizens or land without any outside interference or obstruction. 

This is the core of the question. What gives a country (or, more accurately, a country's government) the right to sovereignty? 

There are at least two things. 

1. International recognition. Part of what gives a government the right to rule is recognition of that right by international organizations and other nations. Consider an example of this: 

The Principality of Sealand is a small micronation off the coast of England. (Check out this news article about a recent development in the nation.) It really only occupies a small island fortress. The "rulers" of this micronation have claimed sovereignty, but this claim has not been recognized by any governments. So the self-proclaimed country is not really a country at all. It has an unrecognized government and, thus, no real power or sovereignty. If the government of England wants to enter the island, they can and will. 

2. Social Contract. We've already discussed how international recognition is important in a country's sovereignty, but recognition of a government by its own citizens is important too. 

Most political theorists argue that a social contract exists between a government and its citizens. This is simply a description of the what a government will offer its citizens in return for their allegiance and loyalty. In the United States, we call this the Constitution. 

Most governments offer their citizens protection, basic services, and a right to be represented in said government. The citizens, in turn, offer the government the right to exist, to make laws, and to levy taxes, among other things.