These titles usually have their origins, in Europe, in the Medieval Period. They are roughly as follows:

Empire- the highest ranked title, from the Latin “imperium” led by an Emperor, Latin “imperator”. Claiming the title of Emperor in a European context from the time of Augustus to at least the seventeenth century was a conscious claim to be successors of the Roman Emperor. With the deposition of Romulus Augustulus in 476, there were no emperors in the west until Charlemagne, who was crowned emperor by the Pope. His successors used the title until their realm split apart into numerous successor kingdoms. Otto I took the title in the tenth century and his successors became the Holy Roman Emperors, claiming to rule three kingdoms of Germany, Burgundy, and Italy. From the reign of Otto until Napoleon, no other monarch in western Europe consistently attempted to claim the title of Emperor, and none were recognized outside their realms (the best example of an attempt were the Christian kings of Leon). Once the Holy Roman Empire was ended by Napoleon, there was a flurry of claims to the title over the nineteenth century—Britain via India, Germany, Austria, and France. Be careful not to be confused by historians speaking of “empires” in a historical context—they often use the term to apply to states which were not seen as empires at the time, like the Angevin Empire.

Kingdom: A king follows an emperor in precedence. In Latin it is “Rex” of a “regnum” or kingdom. Most kings were initially associated with peoples rather than with countries (hence, King of the Franks rather than King of France), but over time the trend was to identify kings as rulers of a particular, named territory. A king usually had a special relationship with the Christian Church of his/her realm. A king might report to an emperor as his vassal, the kings of Bohemia being a famous example. To get around the fact that the emperor technically outranked them, kings began claiming in the later Middle Ages to rule as “emperors in their own realm” without using the title.

Duchy: A duke followed a King in precedence and was literally a “leader” or “dux” of a “Ducatus” or duchy. They came from a military rank in the Roman Army which was adopted by many “barbarian” rulers in the early Middle Ages. By the high Middle Ages, dukes ruled over sizable territories and were commonly found as vassals to a king. A count or viscount might report to a duke as their vassal. Dukes lacked the sacral quality of kings and the special relationship with the Christian Church of their realm. A roughly equivalent rank, Margrave (or Marquess) was used for the equivalent men to dukes who ruled territories on the border of a realm. Their realm was called a March.

Counts: A count is the least in rank of these major titles. In Latin they are “comes” or companions, and that reflects their original status as representatives of a ruler, with a personal relationship. Under Charlemagne and his successors, most of Western Europe was divided into counties. In the days following the breakup of the Carolingian Empire, many counties were subsumed into other counties, or began to report to dukes. Counts could sometimes be just as powerful as dukes, but had a lesser prestige. This lead to many counts—like the counts of Toulouse—finding an old ducal title to claim to better reflect their status. In England, their equivalents were “Earls”.

In general, the biggest “jump” in prestige was between duke and king. A king had a sacral and religious role which was lacked by those of lesser rank. It’s important to remember that ranks of prestige by no means always reflected reality. The fifteenth-century kings of Navarra were far, far weaker than the Dukes of Burgundy. Their respective titles simply reflected the state of affairs that had existed centuries in the past.

Edited to correct a typo about Napoleon and added a clarification about Western Europe.

Interesting question. The United Kingdom is a great example to use to illustrate the definitions, because its territories encompassed all of these terms at one time.

Empire: This is a collection of many different countries, ruled over by a single monarch / state. The British Empire was extensive and spanned large swathes of the world at its peak.

Kingdom: this is more of a statement about the ruler, than the size of the territory. The United Kingdom is a collection of countries (England, Northern Ireland, Scotland and Wales) ruled by a single monarch, hence the United in the name. A Kingdom can also be a single country ruled by a monarch.

Duchy: Again, this is a comment on the ruler of the territory, not the size. It is an area ruled by a Duke. The United Kingdom has Duchies incorporated in its mainland territory, one of which is the Duchy of Cornwall - the lands of the Duke of Cornwall (who is whoever the eldest son of he ruling monarch is, currently Prince Charles - also known as the Prince of Wales and heir apparent to the United Kingdom).

The other is the Duchy of Lancaster, ruled by the Duke of Lancaster, who is whoever the current ruling monarch of the United Kingdom is, so the Duchess today would be Queen Elizabeth II. In both cases, the Duchies are considered the private lands of the holders of these positions (monarch or eldest son of the monarch) but they do not have personal rights to them. This means that the lands belong to the country but provide private wealth to the holders of those positions, but should Queen Elizabeth abdicate her throne, she would no longer actually rule the Duchy of Lancaster, as that would be inherited by the new Monarch, presumed to be Prince Charles. Prince Charles would cease to be the eldest son of the ruling monarch and would therefore cease to receive income from the Duchy of Cornwall. Instead, Cornwall would be gifted to his eldest son. This is much the same idea as the President of the United States having private use of the White House - but none of the Presidents actually own the White House themselves, as it “belongs” to whoever the current President is.

County: This denotes land rules by a count, but today is really an internal division of land within a country, made for administrative purposes. The United Kingdom has over 100 counties.

Not asked, but also interesting:

Land in the United Kingdom is all owned by the reigning monarch, who grants rights to others to use it. This is ultimately how private UK citizens own their homes - they have a leasehold or freehold, which is the last in a line of licences to be granted all the way from the monarch, through various bibles, and down to the private citizen. See also: Subinfeudation.

Hierarchy of titles in the UK, in order of rank: King (Queen), Duke (Duchess) Marquess (Marchioness), Earl (Countess), Viscount (Viscountess), Baron (Baroness), Baronet (Baronetess), Knight, Commoner (private citizen).

Edit: typos. Edits: corrections from comments.

In terms of what polities had these names, and at what times in history, here's a little bit of context for the origins of the terms.

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Emperor, Duke, and Count are all terms that emerge out of the Roman Empire, and come from titles that formerly would have applied to functionaries in the roman empire.

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The Emperor - Imperator - was of course the ruler of the Roman Empire, but just who is an Emperor, and what is an Empire has varied throughout history. While we would apply the term "Empire" retroactively to expansive realms that existed before the Roman Empire, and onto realms in other Non-Roman Cultures. Ultimately, the Roman Emperor was an at least nominally elected military autocrat, and while the position was often hereditary, it was only so if one of the Emperor's sons was a fit leader.

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The other main Emperor to appear in the Middle Ages, and help to define what it meant to be an Emperor, was the Holy Roman Emperor, and similar to his Regular Roman Counterpart, the Holy Roman Emperor was elected, but by a group of Elector Princes, and confirmed by the Pope. What both Emperors - the Basileus and the Kaiser (and eventually, the Tsar) had in common was their claims to Roman Heritage. Historians have liberally applied it outside of its European, and Roman context, but I think if we're strict, those are really the only two people who it's proper to consider Emperors, which is by no means to downplay the power of other people who get called Emperors.

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Dukes and Counts are similarly concepts of Roman Descent, and come from the Dux, and the Comes respectively. Dux was an informal title referring to a military leader of any kind in the Roman Empire, and in later years, would become official, a sort of counterpart to a regional governor - the highest military leader in the government. In Western Europe, which we are principally talking about when we use the word Duke, this distinction collapsed among the Lombards, Franks, and others who took over after the collapse of the Roman Government, but still kept many of their traditions, and Duces, or Dukes, were often regional governors, and the realm's highest officials - they could be chosen as the King, or Emperor's generals in times of war. The Counts get their name from the Comites, the Companions of the Emperor, and are Lords subordinate to another Lord, getting their authority from him.

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King is the only title that doesn't have a Latin Etymology, but it still comes into play in Western thought in the wake of the fall of the Western Roman Emperor. Once you had the fall of Rome, many different Germanic and Gothic nations inherited the realms once ruled by Rome, and their loyalties tended to fall along cultural lines. While it was not by any means universal, it was certainly common to have one King of Franks, or Germans, or Goths, etc.

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So what actually decides who is a Count, a Duke, a King, or an Emperor? It ultimately just comes down to about three things. If you were a subordinate Lord, what honor did your liege grant you? Or what could you get away with styling yourself as, whether you were dependent or independent? And finally, what do historians decide you were? Many people outside of Post Roman Western Europe are still styled as Kings or Emperors, for example, even those conceptions are really specifically Post Roman Western European Ideas. Count and Duke are almost never applied outside of the context of Western Europe, and it's often more accurate to talk about Lordly People from other cultures as Princes, or really, whatever is the culturally equivalent title.

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Ultimately, outside of a Western European, Post-Roman context, terms like Count or Duke don't have much meaning, and are rarely ever used. Kings and Emperors, while terms that are used more broadly, are still terms that apply in a primarily Post-Roman context. A lot of who was a King vs. who was an Archduke ultimately just came down to politics of the secular rulers against the Pope, and one of the only Arch-duchies was created specifically to give the Habsburgs (who created it) leverage against the Prince-Electors.

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I'm not sure in retrospect that I answered exactly the question that you were asking, but I hope that at least helps you understand the way that those titles came to be, why they were used at the time, and why they are sometimes used in retrospect.

Follow up question, why does the United States have counties (as opposed to provinces or prefectures)? Was it just a convenient English term for a "larger-than-city-but-smaller-than-state" subdivision, or was there any greater significance? Was a mayor of a town originally a "count", for instance?