Some of the biggest naval battles in history (at least as far as numbers of active ships and personnel in combat are concerned) took place during the First Punic War, including the Battle of Cape Ecnomus during which seven hundred polyremes carrying almost a third of a million sailors and marines clashed.
One of the most decisive naval battles in the ancient world was the Battle of Actium. Which saw future Emperor Octavian defeat the his rivals, Mark Antony and Cleopatra. As his 400 ships were able to defeat his opponents fewer but larger 250 ships.
Early modern
The Battle of Lepanto, called by Miguel de Cervantes (who served there and lost the usage of an arm to an arquebus shot) "the highest and most memorable occasion witnessed by the centuries", was the largest naval battle in western history since the ancient times and the last featuring rowing ships. It saw a coalition of the Spanish Empire and the Republic of Venice, led by John of Austria, engaging and annihilating the Ottoman Empire's fleet captained by Ali Pasha, which gave the Christians the first great victory over the previously invincible Turk navy. The fleet saw both naval action and ground action from deck to deck, with the Spanish marine infantry working to great effect to capture enemy ships.
Let's just say there's a reason why the Battle of Trafalgar is often cited as the most important naval battle of all time. But Trafalgar was merely Nelson's final and crowning moment of badassery, a fitting end to a career for which 'epic ship on ship action' just happened to be the best description (most badass of all being his action at the Battle of Cape St. Vincent, when the outnumbered British squadron overcame a much larger Spanish fleet, rounded off by Nelson, only a captain at the time, boarding and capturing one battleship, only to use it as a means of boarding and capturing another, giving rise to the expression "Nelson's patent bridge for boarding enemy vessels").
Many of the battles from the Second Anglo-Dutch War, in particular Lowestoft, make Trafalgar look like a skirmish. 212 ships in miles-long battle lines slugging away at each other, including one sequence where flagships Royal Charles and Eendracht got in a one-on-one duel.
Modern
One of the first battles, if not the first battle, between ironclad warships occurred on March 9th, 1862 between the USS Monitor and the CSS Virginia. The slug fest ultimately ended in a draw between the two ironclads as neither ship was able to defeat the other's armor (interestingly, both should have been able to penetrate the other's armor, but errors on both sides prevented this - USS Monitor's cannon wasn't fully tested and thus used charges that turned out to be much less than what they could actually handle, while CSS Virginia was intended to use armor-piercing shells that would have been able to defeat Monitor's armor, but the ship hadn't actually brought any as the odds of fighting an armored warship like Monitor were seen as negligible), but it was clear from that moment on: Wooden Warships were a thing of the past.
The Battle of Lissa in 1866, fought between Austria and Italy, notable as one of the only fleet battles between ironclads, as well as the only instance of a wooden ship of the line engaging ironclads in combat, as the SMS Kaiser engaged three ironclad warships at once, damaged two of them, and escaped with heavy damage, but still afloat.
Another similarly decisive battle was the Battle of Tsushima during the Russo-Japanese War in 1905. Fought between the Imperial Japanese Navy and the Imperial Russian Navy, it was the only decisive battle between steel-hulled battleships and the first and only large-scale battle between pre-dreadnought battleships.
4 Japanese and 8 Russian first-class battleships, as well as 38 cruisers, 28 destroyers, and numerous other vessels, were involved in the battle. After having sailed halfway across the world, the Russian fleet were completely annihilated by the technologically superior and more skilled Japanese fleet, let by Admiral Tōgō Heihachirō from the battleship Mikasa. In some of the most Epic Ship-on-Ship Action of the 20th Century, Tōgō (himself inspired by Nelson) managed to get his fleet to perform a U-turn and 'cross the T' of the Russian battle line, so that he was able to split up the Russian formation. Russia lost nearly their entire fleet, including all 8 of their battleships and all 3 coastal battleships. Japanese losses were just three torpedo boats.
The battle effectively won the Russo-Japanese War, making Japan the chief naval power in East Asia, allowing the expansion of the Japanese Empire, as well as leading to World War One by encouraging the design of dreadnoughts. It also formed Japan's doctrine of Kantai Kessen, or victory through a single decisive battle, which guided their World War Two strategies. Surviving ships include the Japanese flagship Mikasa as well as the Russian protected cruiser Aurora.
The largest engagement between big gun battleships was the Battle of Jutland off the North Sea coasts of Denmark during World War I, when the Royal Grand Fleet and German High Seas Fleet embarked on a campaign to either destroy the other or drive them away from their own shipping lines. The Germans ended up claiming a tactical victory as they sank more ships and lost fewer men, but overall it was a strategic victory for the British, as the German fleet would never be in a position to threaten the British like that again.
Despite the dominance of the aircraft carrier in World War II, there were still plenty of engagements during the war that involved primarily surface ships. Some of the more important ones included:
The Battle of the Denmark Strait — British battleships HMS Hood and Prince of Wales vs. German battleship KMS Bismarck and heavy cruiser Prinz Eugen.
The final fight of the Bismarck, in which shenote Contrary to popular belief the Germans consider(ed) ships female like in most other European languages (despite being gender neutral, a named ship is still referred to as 'she' in German), instead it was Bismarck's captain, Lindemann, who insisted on referring to Bismarck as male, under the belief nothing female merited being given that name. stood alone against the British battleships Rodney and King George V, as well as a substantial force of cruisers, destroyers, and the carrier group centered around Victorious.
Several battles in the Mediterranean between the Royal Navy and the Italian Navy, the largest of which were the battles of Cape Sparviento and Cape Matapan (including one of the longest ranged intentional hits from naval artillery during the Battle of Calabria when HMS Warspite hit Italian battleship Giulio Cesare at roughly 25km, something only roughly equalled by Scharnhorst on HMS Glorious).
The Battle of the Java Sea, in which a force of Japanese cruisers and destroyers wiped out an Allied cruiser force while taking little or no damage in return.
The Naval Battle of Guadalcanal in November 1942, which consisted of two major engagements on separate nights. The second of these, on 15 November 1942, featured one of only two fights between American and Japanese battleships, when USS Washington engaged and fatally damaged HIJMS Kirishima.
The Battle of the North Cape, between the British battleship Duke of York (plus several cruisers and destroyers, including the plucky Norwegian Stord) and the German battlecruiser Scharnhorst.
The Battle of Surigao Strait, one part of the epic Battle of Leyte Gulf in October 1944, included the second meeting of American and Japanese battleships, and the last fight ever between battleships. The Japanese Southern Force, centered around the battleships Yamashiro and Fuso, ran headlong into an expertly laid American trap. The Japanese force was demolished while inflicting almost no losses on their American foes. American naval historians have always considered it poetic justice that five of the six battleships in Surigao Strait that night - Maryland, Pennsylvania, Tennessee, California, and West Virginia — were at Pearl Harbor on December 7, 1941, and were damaged to various degrees during the Japanese attack that day. At Surigao Strait, they got their revenge.
Russians had plenty of experience with naval combat against the Ottoman navy in the Black Sea. One of their greatest admirals, Fyodor Ushakov, pioneered some of the tactics later used by other great naval commanders. Only politics prevented him from being known internationally. In particular, he eschewed the traditional line-of-battle for close engagements and precision maneuvering and firing, often concentrating the fire of several ships on a single target.
In the Modern World its not entirely non-existent. While larger ships use long range missiles they are useless against smaller and faster boats which inevitably lead to gun battles.
The two incidents between North and South Koreas known as Battle of Yeongpyeong in 1999 and 2002 involved almost entirely guns and 1999 incident even included ramming.
during which seven hundred polyremes carrying almost a third of a million sailors and marines clashed.