In order to speak scientifically about politics, it is useful to know the terminology that political scientists use.
State: in political science, "state" means what most people think of when they say "independent country": i.e., France, Italy, Russia, etc. The definition of "state" includes a people permanently living in a territory (not nomads, e.g.), over which a government exercises sovereignty. Thus, the constitutive elements of a state are a people, a territory, a government and sovereignty (some authors don't distinguish between government and sovereignty, so perhaps it would be best to speak of "a government exercising sovereign authority"). Some federations refer to their constituent parts as "states", in order to emphasize their relative importance and independence (e.g., the United States of America, the Federated States of Micronesia). Political scientists tend to use the term "state" to emphasize the governmental aspect of the country: government officials are called "state officials"; a government minister may have the title of "secretary of state", etc.
Sovereignty: full powers over a given territory. Kings and queens were once called "sovereigns" because they had full power over their subjects and lands. Political scientists do not talk of persons having sovereignty, but states. There are two kinds of sovereignty; internal sovereignty generally recognized as being a monopoly on the use of force and external sovereignty which is generally recognized as being control over ones own internationally recognized borders (illegal immigration can be seen as having a negative impact upon a states external sovereignty).
Nation a group of people sharing a common trait that creates a primary political bond. This trait is most often linguistic, but can be religious, racial, cultural, territorial, or a combination.
Nation-State: A state that is generally conceived of as the "homeland" of a given nation. For instance, France is a nation-state of French people.
Government: A system of institutions and organizations that make up the inner functions and processes of a state.
Power: The ability to convince people to do what they otherwise would not have done. This can be done through a variety of means but typically is accomplished through the following: the use of force (coercion), diplomatic means (bargaining, influence et al.) and Ideological control (setting the agenda so that opposing points of view cannot be heard).
Authority: The recognized ability to wield power.
Sovereign authority: The recognized ability to use power to exercise state sovereignty.
Legitimacy: The recognition that someone should have authority. This may or may not imply that the possessor of legitimacy also possesses authority. For instance, if a country is invaded, that country's former government might be exiled and have no real power, but still be recognized by other states as the "legitimate authority" of the country that they were exiled from.
System: A way of organizing as well as interpreting.