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 may be different nations in a given state and nations can exist without a state
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.
In the age of Empire, these were totally different things. A nation was an ethnological term - it referred to a people group. A state, on the other hand, was a governmental authority.
As such, the Austro-Hungarian Empire would have been a single state with mutiple nations - including Austrians, Magyars (Hungarians), Gypsies, and others.
With the breakup of Empires generally, however, the goal came to be that each people group (nation) would have its own state - the nation-state - in which it could pursue its own identity.
These terms seem to be used interchangably - by whom and where? These are complete different things.
The nation - is the sociological phenomenon - the community of people with awareness of common origin, history, language and culture. The religion can also be the unifying factor - like with the Jewish nation before the creation of Israel.
The state - is the legal term of the administrative structure having the power over the given territory.
The state is the territorial entity, the nation - the sociological. The nation can exist without any state; it can be scattered over the whole world and still be the nation. The state can exist without any nation, by ruling over various nations having nothing in common; an example is an empire.
Nation is a way of feeling,thinking and living,it is a psychological condition of the mind which emphasizes the consciousness of unity due to spiritual or other feelings.
The state emphasizes political unity and objectivity which is a condition of law,the state is an enforceable obligatory condition inseparable from all civilized ways of living
A state may be defined as a politically organised body of people inhabiting a defined geographical entity with an organised legitimate government whilst a nation is a group of people with a common race,culture,religion and historical experiences but who may not necessarilly live together in a single territory.
The state is a political community living in a given territory with sovereignty.The nation is a social and psychological community based on identical feeling of common culture and heritage. The term nation-state can be traced in the writings of Machiavelli in the beginning of modern age with the union of fragmented Italy. Maleque
The following defines all 3 terms. To ease reading, I do not use blockquotes, but everything below the line below are verbatim quotations.
Source: Introduction to Politics: First Canadian Edition (2012 1 ed.) by Garner, Ferdinand, et al.
We should also give the terms state and nation some scrutiny. Although they are often used synonymously (or joined together to produce "nation-state"), they actually refer to two quite distinct entities. For our purposes—and those of politics more generally—we
might define the “state as a distinctive political community with its own set of rules and practices, more or less separate from other communities. For the specific purposes of IR [International Relations], “the state” refers to the modern sovereign state, which possess a “legal personality” and is recognized as possessing certain rights and duties. […]
We now turn to the idea of the nation, a term that refers specifically to “a people” as opposed to a formal entity. There is no widely agreed definition of what constitutes “a people”; in general, though, the term generally denote[s] a kind of collective identity that is grounded in a shared history and culture and may or may not lay claim to some kind of political recognition as well as a specific territory. […]
[...] These examples reflect the commonly accepted conflation of state and people that produces the familiar term “nation-state,” which, again, reflects the principle of national self-determination. In principle, though, the matching of state and nation has rarely, if ever, been so neat and unproblematic. There is virtually no state in the world that contains a single, homogeneous nation. Many states are made up of two or more “nations" and even these are not always distinct. [...]
The definitions of "nation" you have been given, are not correct.
It is important to note that nations did not exist before the 19th century. It wouldn't have crossed anyone's mind that they belonged to a specific nation. Ethnic groups certainly did exist, as did tribes and clans, but nations did not.
During the 18th century, philosophers began thinking about human rights. It was no longer a given that serfs should unquestionably obey their masters and that the kings powers were provided to them by God. People began to think that they were more equal than previously thought.
This advancement lead to new societal structures and revolutions. The kings lost their power to rule and were replaced by states. The idea that granted states authority to rule, were Nationalism; the idea that people should be divided into nations.
People in a nation should, ideally, feel "togetherness." This togetherness consists of having common history, race, language, culture and religion. People must also "feel" as a nation, otherwise they are "just" a people and not a nation. A people that becomes a nation experiences a national awakening.
For example, the Nordic countries are among the oldest countries in the world. But it wasn't until the 19th century that they become nations. During this period the "identities" of the Swedish and the Danish nation (and later on also the Norweigan and Finnish) were being hashed out. Writers began to ponder questions like "What does it mean to be Swedish?"
In this way, nations were constructed. For some nations, like the Swedish and the Danish, the existing states matched the new nations. For other nations, such as the Italian, Jewish, Turkish and German ones, the states didn't even exist so wars had to be fought to create them.
Nowadays, the idea of the nation is so ingrained in our minds that we rarely even think about it. That ones "national identity" is at most 200-300 years old is unknown to most.