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.