Compared to the rest of 15th century Western Europe, Italian warfare stands out as having a series unique attributes. Is ten days too much to wait before answering? Nah, it's just u/AlviseFalier getting progressively worse at keeping up with questions!
In the 15th century, Western Europe moved away from the traditional system of raising an army whereby a sovereign would bully/cajole/pay his vassals to fulfill their martial duties; by the end of the 100 Years War the King of France had a core of six thousand knights or so "On Ordinance" (A.K.A. the Royal Payroll) with the ability to conscript up to several tens of thousands more by decree of the king. The Crown of England, which already had a well-developed system of recruiting freemen for military service, further strengthened the royal monopoly on raising troops after the Wars of the Roses.
Italy on the other hand certainly didn't centralize, it went down the opposite road! Indeed, between the 12th and 14th centuries the Italian cities, nominally under the fuzzy suzerainty of whichever German Ruler got himself elected Kaiser, ossified into a half-dozen or so distinct independent polities.
So in the 14th-15th century Italian states were wealthy, numerous, and some (The Milanese, the Veronese, and the Florentines) began floating the idea of conquering their neighbors. However, in the 12th century the Italian Cities had only been able to overthrow Emperor Barbarossa after exasperating him over the course of a decade-long whack-a-mole; the cities could barely muster enough men for their own defense. How would one city assemble the manpower to subjugate another?
The answer came in the form of mercenaries, whom I talked about in the past, here's a summary:
The first professional soldiers in Italy not tied to the traditional Urban Council-Vassals of the Urban Councillors arrangement had actually appeared in the thirteenth century after the Crusades, when Germano-Frankish soldiers returning from the Holy Land were left stranded in Italy. Often little more than roving bandits, they were often rounded up and put to good use during dynastic infighting, and eventually lots of them put themselves in positions of serious power (see, "The Other Norman Conquest" by J.J. Norwich).
Subsequent mercenary companies were more often than not veterans of fighting in France and/or Germany, where dynastic fighting had blurred the lines who precisely was fighting against who, and individual vassals and the forces they commanded opportunistically backed different sides based on plunder, risk/reward, and who seemed to be winning. The first "famous" mercenary companies came from places peripheral to the larger conflicts of the time composed of men far from home with little to loose and everything to gain: Catalans, Basques, and Bretons followed by English, Welsh, and Burgundians. Most early companies were mounted men-at-arms; men who had been born into landowning families, but perhaps stood to inherit little more than a horse and a name. These men were quick to attach themselves to powerful landed lords, offering their service in battle in exchange for lodgings and a place in the court. Following their lords to war, many stayed behind to plunder and offer (or extort!) their services to whomever could offer them the most.
Many of these companies had slowly trickled into Italy in the 14th century, being employed first by the Marquis of Monferrat in northwestern Italy (unsurprisingly near France). Although the Visconti of Milan were also early adopters, as after seizing power in Lombardy they rapidly expanded their holdings.
The later appearance of Italian mercenaries can be attributed to a sort of snowball effect. At the late date of 1377, Alberic of Barbiano established the first Italian-only mercenary company. Alberic himself was a son of the Count of Cuneo, a vassal of the Marquis of Monferrat, and was most probably present when his father was called to arms in the conflict between Monferrat and Savoy. Instead of returning home at war's end, Alberic, then sixteen, joined John Hackwood's White Company, a company of English veterans of the hundred years war employed by the Marquis.
The White Company would go on to be employed by the Republic of Siena, and later the Papacy. When the Pope, in the pocket of the King of France, was convinced to move to Avignon, Hackwood found himself putting down revolts as cities under Papal control tried to assert their independence. Florence, Milan, and Padua also raised armies and attempted to redraw their borders.
After witnessing Hackwood's forces raze the rebellious city of Cesena to the ground, the aforementioned Alberic left the company. Disgusted by the unchecked antics of foreign mercenaries (as well as with the Papacy, a pawn of foreign powers) Alberic pledged his services to Barbanò Visconti of Milan, rounding up the Italian mercenaries who would follow him. Alberic's Company of St. George, as they came to be called, would serve as an incubator for many mercenary captains of the next century.
However, this soon created a system by which in the 15th century war became a kind of "Theatre"; campaigns were waged in a (very short) summer campaigning season, and war became a game of figures and hypotheticals rather than actual confrontation: whichever side could hire the most mercenaries usually won, with individual companies regularly switching sides depending on who was paying better, who was winning, or other criteria. In addition, even when armies were within striking distance few actual pitched battles were fought; maneuver and positioning became exponentially more important, as captains tried to keep their companies intact.
When two armies finally gave battle, the encounter was often short and relatively bloodless; as soon as one side worked out an advantage, the "loser" often attempted to swiftly withdraw in good order (often to the frustration of their employers; "They avoid defeat just as long as they avoid battle" wrote Macchiavelli).
Sieges were more common, especially early in the century. Between the 13th and 15th centuries, the Milanese, Paduans, Veronese and Florentines were almost continuously involved in some kind low-level warfare, often bullying smaller surrounding towns into submitting in order to establish a ring of fortified burghs around their capital. Oftentimes, these sieges were perpetuated by the mercenaries themselves, who needed to be occupied lest they start pillaging; perpetuating a vicious cycle by which larger, richer cities were constantly hounding out weaker ones.
Equally important was the use of diplomacy coupled with the threat of war, as thanks in great part due to the precarious nature of the Italian Lords' rule over their states, in Central Italy especially states like Pisa, Florence, Siena, Modena and Bologna were nominally Republics, and should the members ruling councils detect that the fortunes of the dominant family were waning they could easily oust them and concede defeat without a fight (in Northern Italy, on the other hand, the Italian states were more susceptible to irreplaceable strongmen).
This all changed when, at the turn of the 16th century, Charles the VIII pursued a claim to the throne of Milan. With real soldiers. Who stood to fight real battles. The Italians and their mercenaries were completely unprepared. This set off the Italian wars, whereby the fate of the Italian States would from then on be decided outside the Peninsula.
If you'd like to read more about the decline of Mercenary Warfare in Europe at the hand of professional armies of Early Modern States do pick up "War and Society in Renaissance Europe, 1450-1520" by John Rigby Hale (Johns Hopkins University Press, 1986). If you have a JStor account, Micheal Mallet's paper "Diplomacy and War in Later Fifteenth Century Italy" could also be helpful.