When did aircraft carriers eclipse battleships as the "King of the seas"?

The predecessor of the aircraft carriers were the seaplane carriers. UK was the first country that used in naval combat with great success (battle of Jutland). So was the success of that warship that UK improved them into an aircraft carrier. HMS Argus was the first aircraft used by UK in 1918.

As other have said, the change became appreciated during WWII.

An important prerequisite for the change was the advent of aircraft engines in the 1000hp class, which meant that single-engine aircraft that could fit on carriers could also carry bombs and torpedoes powerful enough to be a serious danger to large warships.

Such engines started to come into service in the mid-1930s, which was only about one battleship building cycle before the outbreak of WWII. It's thus easy to see why the change wasn't appreciated until ships had been sunk.

The Japanese sank the Royal Navies greatest and one of their most sophisticated Battle Ships HMS Prince of Wales and its older escort Cruiser HMS Repulse off the coast of Malaysia on 10th December 1941 using land based bombers and torpedo planes. The British ships had no air cover and were sunk relatively easily. These were the first ever capital ships sank purely using aircraft and the Allies realised a handful of torpedo planes sinking the Prince of Wales meant the era of the Battleship was over. Air power was everything and the Carrier was King.

https://en.wikipedia.org/wiki/Sinking_of_Prince_of_Wales_and_Repulse

See Z-Force mentioned in the answer above, Z-Force was HMS Prince of Wales and its escorts.

The question could be divided in two parts: First, when did they loose the title of King of The Sea to Aircraft Carrier, then when did the naval strategy change? And of course, all of that should be considered in respect of each country's naval system.

Germany

The Germans aimed during WW1 to be challenger of the seas. They used battleships against UK's battleships and they failed. But aircraft carrier was a concept by the time. During WW2, the Germans aimed to attack trade and challenge Allied escorts with battlecruisers and fast battleships. The Bismarck/Tirpitz were of them. When the Bismarck was sunk by the British, with great help of the Ark Royal carrier, the Germans ceased to consider battleships as King of the Sea. They still used them in their naval stategy, but with more caution in face of a carrier. Eventually the Tirpitz was sunk by land-based aircraft and the defeat ceased the German naval strategy.

UK:

The British had a strong prescience of the capacities of aircraft carriers. But they also had strong traditions, so they aligned both capabilities. The use of battleships in fight remains very prominent until the defeat of the Z force to massive land-based air attacks. This was the end of battleships as King of the Sea, because initally the Z Force was supposed to save Malaysia from the Japanese (please note that the Z Force was supposed to be supported by a carrier, but losses in the Mediteranean Sea hampered this strategy). But the naval strategy continue to put an emphasis on both means, mainly because:

• Battleships were still useful against German battlecruisers/battleships (see the Scharnorst)
• Battleships still provided support to land attack
• There was battleships in the inventory, so the British used them because they were available. On the contrary, they were carriers but losses of carrier did not help to use them all the time
• Japanese navy was mainly fought by the Americans and not the British

Italy:

Italy mainly based its navy, the Regia Marina, on battleships. The attack of Taranto reduced their availability. But the turning point was the battle of the Matapan, a fight during which one British aircraft carrier compensate the Italian advantage in naval gunnery. Naval strategy changed: the Italians still used battleships (and heavy cruisers), but under the protection of land based air power. This reduced their ability to intercept with naval forces British convoys. Overall, after Matapan the Italians did not consider the battleships as King of the Sea anymore.

USA:

Let's change the part of the world we're considering: The USA mainly fought other ships (except submarines) in the Pacific Ocean. During thoses fights, battleships were involved in various tactical situations. The first major event, of course, is Pearl Harbour: by destroying or damaging most of battleships, this Japanese attack force the Americans to center their strategy on aircraft carriers. BUT it does not mean battleships were diminished: the Americans mainly faced Japanese carriers with their own carriers so it did not mean anything during the battles of Coral sea and Midway. Later however, the naval strategy changed progressively during Guadalcanal fights and later: the battleships were centered on the role of night fighting, while in the Center Pacific they escorted carriers that led the strikes on the Japanese fleet: this was a strategy change Eventually, after the big surface battle of the Philippine sea, aircraft carriers had proven central but battlehsips were still useful and powerful, and the Americans had not yet won a major battle of carrier-based aircraft against battleships: this came in 1945 with the sinking of the Yamato: the King of the Ocean had fallen.

Japan:

Of course, the evolutions for Japan are linked to the ones for Americans. There were some differences of spirit, though, but overall the same events triggered the same changes. Only point: with the early loss of carriers the Japanese had to turn to a aircraft-carrier central strategy to a battleship strategy in 1944, that failed with the turning point mentionned above.

Battleships became obsolete with the rise of air power; this was clear to General Billy Mitchell at the end of WW I.

A 2,000 lb. bomb "near-miss" severely damages Ostfriesland at the stern hull plates (1921)

A series of tests were conducted between 1921 and 1925 which conclusively showed that battleships could be severely damaged or sunk by means of aerial bombing. It took time, and repetition, for the lessons to "sink in", but they did, and this lead to the creation of the air craft carriers which played such a prominent role in WW II.

The balance is found in "cost to build and operate" vs "cost to destroy"; air power is cheaper and more flexible than the battle ship. Note that coastal defenses in the form coastal artillery were also rendered obsolete by air power.

Sea power is not directly about which ship type can beat up which other ship type; it's a matter of being able to run one's merchant ships in an area and preventing the other side from running merchant ships.

In WWII, it was difficult for surface ships to protect a convoy from air attack, as we can see from the 1942 attempts to relieve the siege of Malta. Moreover, naval powers were hesitant to risk battleships in the face of enemy air power, particularly after the British lost a modern battleship and older battlecruiser to Japanese air attack off Malaya shortly after Pearl Harbor.

The questions, then, were whether aircraft carriers could defend themselves against battleship attack and whether aircraft carriers could protect shipping against battleships.

In WWII, carriers were generally able to not be sunk by battleships or other surface ships, unless badly handled like HMS Glorious off Norway. Their aircraft gave them warning of enemy forces, and their generally high speed enabled them to avoid combat. Therefore, they could stay in an area despite enemy battleship forces. (This was not always the case; in the 1930s, when carrier aircraft had much less range, US battleships "sank" US carriers a few times in the annual Fleet Problems.)

Their ability to defend shipping against battleships was a bit less certain. Normally, heavy air attacks would deter battleships from proceeding. After all, there weren't many battleships around, and they were expensive. If the enemy was determined to force his way through, some surface ships would likely make it through. This was shown by RN cruisers and destroyers around Crete and the Japanese Center Force in the Battle of Leyte Gulf, both of which carried out their missions despite serious losses. No naval power could afford to lose battleships with each mission, unless the enemy missions were extremely critical (the Battle of Leyte Gulf was an attempt to stop the first US landings in the Philippines).

Through the course of World War 2; in September 1939 battleships were still considered more important than aircraft carriers, and by VJ Day carriers were considered utterly decisive at sea. This process took a good few steps, however, and really examining it in detail could take an entire book.

You also need to consider the application of carrier air power to two main problems:

1) the decisive battle

2) the guerre de course, or convoy war.

So, first let's look at decisive battle. At the beginning of WW2, it was by no means taken as read that unsupported aircraft could sink a battleship; so while a carrier could do extremely well at finding a battleship, and keep a safe distance from her, sinking her was another matter. And the carrier could hardly replace the battleship if a carrier couldn't even sink a battleship. As a result, any would-be naval power had to have battleships in order to counter enemy battleships, whereas those who expected their navy to operate in range of land-based air support (Italian, German) did not need aircraft carriers. This assumption had to be overcome by experience, and there were several battles which made that example, and then rubbed it in the noses of the gun-deck admirals.

At Taranto in November 1940, the Royal Navy sank an Italian battleship in harbour and damaged two more, for the loss of two obsolescent torpedo bombers; so battleships were proven to be vulnerable to carriers when unable to manoeuvre freely.

In the chase of the Bismarck, the carrier aircraft did not sink Bismarck - however, they did damage her rudder, enabling her to be caught and destroyed by battleships.

Then we have the fate of Force Z - a surface-force only, consisting of a battleship, a battlecruiser and a few destroyers sunk by Japanese air attack. These were not carrier aircraft - the Japanese carrier force were somewhat busy at Pearl Harbor - but the crucial difference with Force Z, which differentiates it from Taranto and Pearl Harbor, is that Force Z was sunk while under sail, showing that any naval force which went into reach of enemy aircraft without its own air cover would be sunk.

Having already mentioned the parallel, I would be remiss to not cover Pearl Harbor - this is essentially Taranto writ large. It showed that if your enemy has a sufficiently powerful carrier arm, your fleet is not safe in harbour anywhere, not even across an ocean.

Then at Midway the combat between the American and Japanese carriers was decisive, without the Japanese battleships becoming engaged.

So, enough about decisive battle - what about the guerre de course, conducted primarily in WW2 by German U-boats?

Well, protecting a convoy against U-boats was helped hugely by air cover. Spotting the submarines in the first place, guiding in escorts to kill them or killing them directly, routing convoys around them (and providing excuses to use ULTRA intercepts to route convoys around them), forcing submarines to submerge and lose a chance to make an attack...the uses of aircraft in the Battle of the Atlantic were many and varied. However, at the start of the war there was a gap in the mid-Atlantic where land-based aircraft could not reach a convoy to protect it. Some desperate measures were used to help close the gap, including CAM ships (Catapult Aircraft Merchantmen which carried a single Hurricane on a catapult; this could be launched once, and at the end of the flight the pilot had to ditch the plane and bail out), the slightly more capable MACs (Merchant Aircraft Carrier) which carried about 80% of a normal cargo and had three or four aircraft embarked which could be recovered after a flight. These were stop gap solutions - as the war went on, they were gradually replaced with escort carriers, which were generally akin to MACs in that they were based on merchant hulls, with the difference that they were full conversions no longer carrying any cargo and could carry perhaps 20 aircraft - they were still too slow to operate with a fleet.

There were many aspects to the outcome of the Battle of the Atlantic, but unlike decisive battle where the battleship admirals resisted fiercely the supplanting of their beloved big-gun ships, the usefulness of air power to this aspect of naval warfare was little doubted, and the frantic attempts to close the Air Gap in the Atlantic show this quite clearly.