What is the difference between a nation and a state?
These terms seem to be used interchangably - what is the difference between them?
Additionally, why is it that a "state" appears to be just an entity of a federal government in certain contexts - e.g., "the United States?"
Additionally, how does this apply (if at all) to the definition of "state" as used by the United States of America?
There are three different things to define here:
State: "A state is an organized community living under a unified political system, the government" (Wiki definition).
This is basically just a community (usually in a specified territory) that was ruled by a specific government.
It may or may not have been sovereign.
Nation: A nation may refer to a community of people who share a common language, culture, ethnicity, descent, or history (Wiki).
Note that a nation has no required geographical tie-in (as an extreme example, consider the nation of Roma, or post-Diaspora-pre-modern-Israel Jews). But they must/should, as a rule, share history, culture and language (never thought I'd quote Stalin on a Politics.SE :)
The idea of a nation and a state being the same thing ("Nation-state") is fairly new in modern politics (it came about as one of the consequences/results of Peace of Westphalia, which ended the 30-year-war in Europe, when the concept of "Westphalian sovereignty" was introduced).
Before that, a vast majority of people did not - per se - had a firm notion of a "nation", at least in Europe. Your loyalties were either to your immediate locale (village, town, clan), or to your hierarchical ruler (feudal lord, usually, and ultimately whichever prince/Emperor ruled the whole territory of the state).
But you didn't consider the territory ruled by that Emperor to be "your" state - the fact that they shared the ultimate liege lord was irrelevant both practically, AND philosophically/culturally.
Also, as alluded to by @Affable's answer, most of the states through history did NOT match the notional nations geographically, and many crystallized as nations much later than 1648. For example, Russia at first had a gazillion city-centered small states. Then, when they were unified under a central government, they fairly promptly ate up the territories of OTHER nations and turned into transnational empire, which they still in effect are. Germany wasn't really unified as a nation till either Congress of Vienna in 1815 or even Bismarckian unification in mid-19th century. Italy was a collection of warring states till 1861 or so. Islamic empires/khaliphates were transnational and Arab Nationalism didn't really exist till 20th century, after Ottoman Empire was gone.
 - There were earlier attempts and ideas to create a nation-state - for example, I would consider Alexander the Great's efforts to be such, since he tried to unify the culture of his empire (and definitely succeeded doing so in the scope of Greek Peninsula itself). I'm not sure how to classify Rome - they tried to also strongly unify the culture of their empire, and had a Latin as common language.
Rome was an empire, which by necessity must be both multi-national and multi-cultural... It can even include "nominally" sovereign states, however the economic system is always one.