A king is generally understood to be a sovereign ruler, answerable only to God.

Duke, from the Latin Doux, was a title initially referring to a military leader which eventually came to refer to the ruler of the largest subdivisions of a feudal Kingdom. France, for example, was traditionally divided between Burgundy, Normandy, and Aquitaine (plus the Archbishop of Reims who for all intents and purposes also ruled a duchy). A duke then subdivided his duchy into counties ruled by counts; The Duke of Normandy, for example, split up his Duchy in the counties of Rouen, Meulan, Evereux, Eu, Contentin, Cau, Bayeux, Avranche, Aumale, and Alençon. The Counts, in turn, would split up their counties among Barons and/or Viscounts (Barons with special privileges). Of course, as we all know with feudalism there are all a manner of exceptions: Ssme counties, like the County of Toulouse, didn't even belong to a Duchy, being directly under the rule of the King (and the King later ruled it directly as his own possession from the thirteenth century on). Further, several Counts under the direct rule of the King who had either by purchasing or marrying (or even sometimes war) acquired large, near-Duchy sized territories were elevated by the King to the rank of Duke (Bourbon, Orleans, Anjou, Berry, Angoulême, Alençon, Touraine).

Some Dukes did become kings; after the title King of Italy had been usurped by Charlemagne, a rump state was formed in Southern Italy centered around the Duchy of Benevento which did not recognize the Frankish Emperor. Over two hundred years and lots of events later the ruler of Southern Italy, Roger II (who primarily went by the titles Count of Sicily and Duke of Puglia, two titles seized from the Arab Caliphate and Byzantine Empire, respectively) was able to convince pope Anacletus II to elevate him to King (the logic being, the Pope is Christ's vicar on earth and decides when a ruler is answerable only to God, plus there was a whole lot of confusion as to who Roger was answerable to otherwise).

There were a number of reasons why some independent Dukes, especially in Italy, did not adopt the title "King." In Milan, for example, because the ruler was generally understood to hold the office at the pleasure of the Signoria, or Noble Council, (and not as vassal of the King of Italy, who kind of evaporated) they were called Signore di Milano even if they ruled a Duchy-sized territory as rich as a Kingdom. This useful caveat would allow members of the ruling Visconti dynasty to assume the title even if they also held the office of Archbishop (a useful way to maintain power when the previous ruler did not have a suitable heir, as the archbishop presided over the council). The title Duca di Milano was only adopted by Gian Galeazzo Visconti in 1395 mostly for protocol reasons when he was trying to marry his daughter to the Dauphin of France. He had to file a whole lot of paperwork with the King of the Germans Wenceslaus (who had a debatable mandate to give it to him the title in the first place, having never been crowned Emperor by the Pope).

Emperor, from the Latin Imperator, was at first title used by rulers claiming to be successors of the Roman Emperors (but more practically, successors of Charlemagne) the necessary condition being that they be crowned by the Pope. However after Napoleon the title came to refer to any king ruling over multiple kingdoms, ergo Queen Victoria gets crowned Empress of India, and Victor Emmanuel III adopted the title Re e Imperatore after fascist Italy occupied Albania and Abyssinia and he appropriated their regal titles.