The terms duke and count come from Latin, and originate as military titles (dux and comes, respectively). I know a lot more about the evolutional of ducal titles so I'll focus on that, but from the medieval period on they are functionally the same, just on a different scale. In the late Imperial period, the title of dux was linked with a province, and signified the highest military authority in the province, eventually being able to act independently even of the governor of their province. With the Germanic migrations and the erosion of the Roman empire, the Roman provinces at first largely retained their borders, and the titles continued to be used; Henri Pirenne, in Mohammed and Charlemagne, argues that the Germanic rulers saw themselves as a continuation, rather than replacement, of the Roman empire, even after their practical and official fealty to Rome ended. The duces of Germanic western Europe were also landowners, and this, in combination with the military power they held by virtue of their title, meant that they soon became the de facto and then de jure authority over the land their title corresponded to.

In the early medieval period, the majority of nobles in Europe lived in the territories they held. Over the next thousand or so years, the consolidation of titles by noble families, through marriage, bequethment by sovereigns, and military conquest, meant that one person could hold a collection of titles to lands theoretically all over Europe. This caused a number of problems, a notable example being the Norman kings of England, who, having first held, and continuing to do so after the conquest, the title of Duke of Normandy, were legally vassals of the French king, a situation that lead to centuries of political and often military conflict. Another result was that these nobles had to either choose one of their domains to rule from, or travel constantly between all their holdings. In situations where this was not tenable in the longer term, many nobles would divide their distant holdings among a collection of heirs in their will, or bequeath some of their holdings to vassals to rule in their name during their own lifetime, so that they could be ruled by a lord residing locally, a situation that was noted later by Machiavelli in the Prince to contribute greatly to stability, and be much more likely to result in the continuation of control over that domain. Another solution was to grant a family member the authority to rule a fief in the lord's name without transferring the title; this could be dangerous, as an ambitious and unscrupulous person could attempt to usurp that domain for themselves, or at least use their authority to enrich themselves while withholding the fruits of the fief from its titular lord.

As the Germanic kingdoms of Western Europe evolved, the kings, originally elected by the dukes from among their number, became hereditary. At the same time, the system of feudalism was evolving into what we think of today when we hear the term. A noble ruled a portion of a kingdom in the king's name, earning for themselves the bulk of the profits those lands produced, and in return, owed military service in the form of a certain number of soldiers for a certain period of time per year.

Higher titles (duke was usually the highest title under the monarch, with count below that, and baron below that) usually came with larger fiefs, which were often divided into smaller domains; so, for instance, a duchy might be divided into a small collection of counties and baronies, with some of those counties possibly being partially made up of baronies themselves, and only a part of the land being directly under administration by the duke himself. The less of their land the duke was directly responsible for, the easier it was to hold that domain in absentia, as subordinate lords who were present themselves could maintain stability within their own fiefs, and if they owed military service to their lord as part of their vassalage, be called upon to suppress any uprisings in the higher lord's own direct holdings.

Even in the late medieval and into the early modern period, many lords resided in their holdings; the Sforza Dukes of Milan are examples. In England though, most titles have become largely symbolic; the Prince of Wales, for instance, has been since the 13th century the title held by the English heir apparent. As titles have lost their civic roles, the importance of residing in the associated fiefdom has completely disappeared.

Please forgive any spelling errors; I wrote this on a smartphone touchscreen.

Edit: I always wondered back when I read those books, with all the intermarriages and houses being wiped out, how every lord held exactly one land-bearing title and no family had more than one branch.

Edit 2: if I recall correctly, 'earl' is the English equivalent of continental 'count'. Earl comes from Nordic jarl, a title used in Scandinavia, and by extension, in Britain during the Danelaw and the Viking domains in England, Scotland, and Ireland. Perhaps someone else can elaborate on how earl came to be a subordinate title to duke, a title which gained its Germanic feudal connotations in Britain under the Normans.