Emperor Tigerstar recently made a video (on 01 Nov 2017) about what country, nation, state, and other such words mean. The debate was analyzed from a more historical point of view, and I thought it would be nice to look at the issue linguistically, with commentary, to shine the light at another angle. This page is aimed at English learners, but secondarily to native speakers interested in etymology and semantics.
Don't even bother looking for a consistent definition of empire - these definitions are typically inconsistent prescriptivist nonsense. Typically, whether a country is an empire or not depends on what definition you mean. I list the preceding three definitions, roughly in descending order of how common they are. Note that the third definition I list is not limited to countries! For example, this sentence, slightly reworded from Wiktionary:
McDonald's is a fast food empire.
Nobody is insinuating that McDonald's is some sort of nation wielding an army or something. This sentence simply states that McDonald's dominates the fast-food market.
Likewise, I can say the following:
Iraq is the latest state of the American Empire.
The United States is clearly not ruled by an emperor, rather, the one saying this would be implying that the United States has such imperialistic tendencies. The speaker would also likely have very strong views on the American occupation of Iraq!
And, finally, edited from the Wikipedia article on the Byzantine Empire:
The Byzantine Empire was the continuation of the Roman Empire in the East during Late Antiquity and the Middle Ages.
This follows the first definition, and the most common. The Byzantine Empire was, in fact, ruled by an emperor.
Etymologically, empire comes from Latin via French, as many words do. In Latin, the word had a much broader definition. In addition to the first definition, imperium also, roughly, meant "control", "law", or even "order" (ie. a command). Imperium comes from impero, a verb meaning command, which is a prefixed form of the verb paro, meaning prepare or arrange. Paro ultimately comes from Proto-Indo-European *per-, meaning bring forth. *per- also brings us native English words such as fare and ferry.
Usually, when referring to a 'sovereign state', people use the word 'country', unless they're being uptight pedants, in which case they use confusing and contradictory language to demonstrate how smart they are, or something. This word, thankfully, nearly universally refers to the first definition, though you often see country used in the second definition, like in country music. Occasionally, certain provinces in countries are themselves called countries, for example, Basque Country, a province of Spain.
Etymologically, country comes from Latin via French, yet again. It is abbreviated from terra contrata, or 'the land opposite'. Contrata is a Vulgar Latin word from Latin contra, meaning 'opposite' or 'against'. Contra was formed from Latin cum 'with', by analogy with intra 'within'. Cum comes from Proto-Indo-European *kóm meaning 'with' or 'next to'.
Both are used equally often. Provinces of the United States and Mexico are called states by their inhabitants. Outside this usage, state typically refers to a country, eg. sovereign state or nation-state.
Etymologically, state comes from Latin via French, no surprise again. In Latin, status is the perfect passive participle of sto 'stand, stay' and literally means 'fixed, set, regular'. Sto comes from Proto-Indo-European *steh2- meaning 'stand up'. From this root we get native English words like stand, stay, stead, stow, and stud.
Territory, thankfully, is one of the least ambiguous words listed by Tigerstar. All three of these are used about as often. Often times, like territories of the United States and Canada, territories have fewer rights than states (US) or provinces (CAN). In the United States, territories cannot vote in presidential elections, but most parties allow them to vote in the presidential primaries for some reason. Territories of the United States get congressmen like states do, except these congressmen are completely powerless and have no voting power, which is really retarded, but ok whatever. From the best I can understand, in Canada, the difference is that territories have no inherent rights and any rights given to them are given by the federal government. I'm sure other countries have other definitions but I'm not gonna list them all here, two is enough.
Etymologically, territory comes from Latin... directly! No French intermediary. In Latin, territorium comes from terra 'earth' + -tor '-er' + -ium [noun-forming suffix]. Territorium in turn comes from Proto-Indo-European *ters- 'dry', which gives us native English nouns like thirst.
Tigerstar lists only the second definition, claiming the United States is therefore not a nation, but in actual usage most would agree it is a nation. Nation is usually used in the first sense. The second sense is a more technical one.
Etymologically, nation comes from Latin via French. In Latin, natio only possesses the second meaning, and has a more literal meaning of 'birth', and comes from the verb nascor 'be born'. Nascor comes from Proto-Indo-European *genh2- meaning 'give birth'. From this root we get native English words like kin and king.
The closest modern equivalent would be the EU. The HRE - especially later on - was an extremely loose collection of states. These states were more or less independent. It is most helpful to think of it as a confederation.
It fits the first, third, and arguably second definition of empire, so it is definitely an empire. It doesn't faithfully capture any of the definitions of country, state, or territory, so it is none of those. Maybe you could argue it was a state. The only non-German territories the HRE controlled were Bohemia and Silesia, but even those were very German. The HRE was an empire, but not country, state, nor territory. Maybe you could call it a nation, but that's a stretch. Wikipedia categorizes it as a 'multi-ethnic complex of territories', an extremely neutered but correct description.