Italy
Overview
Italy in the Middle Ages was not Italy. Oh, the peninsula was there, true enough, but no nation existed. Instead, the peninsula was divided into dozens upon dozens of principalities and territories, all vying with one another for land and power.
We can simplify the politics by recognizing three main divisions. These were, indeed, recognized by the time and long pre-dated the Renaissance.
The South
In the southern end of the peninsula were two kingdoms (which sometimes were joined into one): Sicily and Naples. The Kingdom of Sicily was just the island. The Kingdom of Naples (for a while called the Kingdom of Trinacria) consisted of the territories from Naples to the end of the peninsula. For our period, these kingdoms were always ruled by outsiders—by French or Aragonese.
Papal States
In the center was a collection of small powers known as the Papal States. The popes didn't always have much power over these, at least not until the mid-15th century. Some made various moves toward independence, which caused states to the north or south to be inclined to meddle here. Between that and incessant internal rebellions, the popes had their hands full with this area. Control over central Italy, and defense of it against outsiders, was at the forefront of papal policy, even when the popes were far away in Avignon.
Northern Italy
In the north were the great Italian cities such as Venice, Florence and Milan. Much of this area, especially in the Po River valley, was in theory ruled by the Holy Roman Emperor and was called the Kingdom of Italy, a title that was inherited from the days when Charlemagne conquered the Lombards.
From time to time, especially in the early 14th century, a German emperor would make a try at actually ruling here, but none were successful. The result was an unending series of wars throughout our period, except for a quiet stretch in the later 15th century.
Themes
Economic Background
Italian states—whether city-states or principalities—struggled to control their finances the same as did the states north of the Alps. It's true, however, that many of the Italian states did a much better job, both of raising money and of managing it. As much as from commerce, the wealth of Italian cities came from their ability to manage that wealth once they'd acquired it. Even so, most governments were constantly in financial need, and this conditioned political decisions, especially when it came to the conduct of war. You'll see numerous examples of this.
Cultural Background
The religious crises that wracked Europe hit even harder in Italy. The absence of the popes, the papal schism, the challenges to orthodoxy, all struck home not only in the Papal States but also particularly in northern Italy. And, of course, these centuries are home to the famous Italian Renaissance. While mostly art and literature seems to play out on a side stage, relegated to an "and also" chapter in most textbooks, it was of more central concern to the people at the time. Not only did cities and lords seek out Renaissance painters, sculptors, architects and writers, but those same artists chronicled and depicted Italian life in a detail not found in earlier centuries. This naturally tends to color our perceptions of Italian society.
Political Themes
The main conflicts in Italy derived from the tensions within and between the three areas of south, center and north. Naples was always looking to expand northward. Sometimes its kings were willing to work with the popes and aim their ambitions at northern Italy; other times, they were in conflict with the popes.
The papacy was always concerned to effect a balance of power in order to preserve the Papal States. This meant that they might in one year ally with Venice to cow Milan, but in another year might call upon Florence to humble Venice. And they were not afraid to call upon the French or Spanish to settle matters, either.
In the north the politics were even more complex, but the great cities, too, were concerned not to be ruled by outsiders. For this reason, they might seek imperial assistance as leverage in a local quarrel, but would resist any emperor who attempted actually to rule them. Likewise, they tended to resist any pope who interfered too actively. And, like the popes, they did not hesitate to call upon various non-Italian powers to improve their position, always expecting those powers to go home again when the job was done.