How were the wars in 15th or 16th century Europe fought?

This is a very broad question; I saw your previous question about medieval warfare, and I encourage you to narrow down your answers further in the future. For this question specifically, I'll point out that the fifteenth and sixteenth centuries are not typically grouped together, as the development of gunpowder weapons increasingly came to define the 16th century much moreso than the 15th, though medieval historians do argue that the situation was more complex and there was more continuity.

Still, this is as good an opportunity as any to discuss warfare in a grand strategic theatre that really defined by continuity through the fifteenth and sixteenth centuries. The Mediterranean Sea was host to a great power struggle through the late medieval and early modern period, seeing the rise of the Ottoman Empire in the East and the Spanish Empire in the West. In the middle was Venice; on the one hand subject to intermittent depredations by the Turks, and on the other dependent on trade routes under their control.

War on the 15th-16th century Mediterranean was profoundly amphibious; developments on land and sea had strong reciprocal impacts. If you couldn't reach a city with your land army, you could still take it with your fleet. Even if your fleet was defeated, your land army can still defeat the enemy navy by taking the key ports they depended on.

The key weapon of this theatre was the galley. Able to prowl through shallow coastal waters, hide in inlets, dash up to lumbering sailing ships for boarding, dash away to escape a stronger enemy, and land on beaches, the galley was a highly versatile weapon of war and piracy. While our attention is usually focused on the great battles and sieges of the 'big war', what really defined warfare in the Mediterranean was an ongoing 'little war' of economic attrition. Individual ships or small squadrons raided shipping and made inland raids, plundering villages, and taking slaves.

However, the galley paid for this tactical performance with some strategic limitations. With their large crews and narrow, shallow hulls, galleys were only capable of relatively short voyages from friendly ports, as they did not have much room in the ship for carrying large quantities of supplies. In rough terms, the galleys range was about 20 days of sailing, which meant that its operational radius was about 10 days. Thus at its core, warfare revolved around the control of key fortified ports, which served as bases for continual raiding.

Warfare was also a seasonal activity; the wet Mediterranean winters mean storms on the sea which hinder military activity. That's not to say the 'little war' ground to a standstill from November through February, or even that major campaigns were not undertaken in this timeframe, but they were less intense/frequent, respectively. Charles V's expedition to take Algiers was launched quite late in the campaigning season, so it's not terribly surprising that his invasion force took heavy losses due to storms. After the Battle of Lepanto on 7 October 1571, there was little time to exploit the destruction of the Ottoman navy.

Another limitation was the availability of manpower. Running a war galley was not a simple task, requiring skilled sailors, deck fighters, and to a degree oarsmen. For much of this period, states lacked the institutional muscle to train people in the sense familiar to us today -taking human raw material and teaching them unfamiliar skills- and instead relied on men who learned their skills through a lifetime of experience. The Ottomans relied on a corps of bowmen who developed their skills from childhood; the Venetians recruited their deck fighters from the Dalmatian coast, in what's now Croatia and Albania. When they took heavy losses, or their recruiting grounds were conquered, they were difficult to replace. You can see the importance of manpower in the costs associated with operating a galley; it cost three times as much to crew one for a year than it did to build it from the ground up.

In terms of tactics, the war galley was a profoundly offensive tool, designed for boarding. All of a galley's fighting power was concentrated forward; the deck fighters' platform was at the bow of the ship, the long pointed spur at the bow served as a boarding bridge, and once they were armed with artillery, all of it pointed forward, like the armament of a jet fighter. In ship to ship combat, it was much better to board than to be boarded, and even better if you could attack from the flank. Galleys did not carry enough artillery to sink their enemy in a single volley, and given the long reloading process, they were unlikely to get a second volley off before swords crossed. As such, galley captains preferred to reserve their fire until the last moment, to ensure the maximum impact. Even if they could destroy the enemy vessel by gunfire, it was usually preferable to try to capture it.

However, galley to galley combat was not the galley's only use. As mentioned earlier, ('Big') warfare revolved around the capture and occupation of fortified harbors; galleys were useful for both attacking and defending such places. Thanks to their shallow draft, maneuverability, and heavy guns, they could serve as close range siege batteries against seaside fortresses, or they could feed reinforcements and supplies into a besieged stronghold and evacuate wounded. Thanks to a timely relief squadron, the city of Famagusta in Cyprus was able to drag out the Ottoman conquest of the island through the winter of 1570 into the next year, leading to heavy losses, which some believe resulted in their total destruction at the Battle of Lepanto later that year.

What galleys typically didn't do, though, were the things we most commonly think of when we talk about naval warfare: fleet battles and blockades. Because of the galley's logistical limitations, maintaining a blockade over a wide stretch of sea was for the most part impossible. This period saw a paucity of general engagements among battle fleets, as such an engagement was difficult to bring about even if it was desirable. Thanks to their shallow drafts, a fleet that felt it was at a disadvantage for a possible battle could backwater all the way up to shore, guns pointed to the sea. This was a very strong position for a defending fleet, as troops could more easily be shifted from one sector to another, the crews could be kept at a safe distance, the firing platform was more stable, and land troops could be brought up for support when available. The stronger fleet would be unable to maintain contact with the smaller fleet it sought to battle, as it would eventually have to break off to resupply. Outnumbered 5-2 against a large Ottoman fleet, Andrea Doria, Genoese captain in Spanish service, was able to escape with the loss of about seven of his 38 ships.

During the late sixteenth century, a few factors led to the decline of traditional coastal warfare in the Mediterranean. After developing new rowing techniques that allowed ships to carry larger crews of oarsmen and deck fighters, the logistical burden of campaigning with a galley fleet increased. Moreover, the price of meeting this burden increased dramatically; the biggest single expense of the galley captain, biscuit for the crew, quadrupled in price over the course of the sixteenth century. At the same time, the Atlantic states like England and the Netherlands developed new cannon casting techniques, and armed their ships with more cast-iron cannons. Between increased costs, reduced strategic range and utility, detente between the great powers, and increasingly heavily armed sail-driven roundships, traditional galley warfare had ever less of a place in the world.