I've previously described the use of carriers in the Atlantic, but this only covers half of your question. While the Royal Navy was the only European power to use carriers in force, it's hard to say that the other European powers ignored the carrier. In 1939, the French were operating the carrier Bearn, converted from the hull of a cancelled battleship, with two more purpose-built ships under construction. The construction of the German carrier Graf Zeppelin was well underway, though the ship would never be completed. Italy had not constructed a carrier, but their designers had produced several designs for new carriers, or for converting existing ships to carriers.
As I've previously covered in several answers, the Royal Navy were enthusiastic users of aircraft carriers. They saw them as a vital and central part of the fleet, able to slow and damage an enemy battlefleet or protect the RN's own. However, British carriers were held back by poor planning decisions, which resulted from the fact that British naval aviation had been split off from the RN and handed over to the RAF in 1917. The RAF was completely uninterested in naval aviation. They underfunded naval aviation and hived off development of naval aircraft to second-rate companies. They also suborned naval aviation doctrine to the ideas of strategic bombing, hampering the development of things like dive bombers and long range fighters that would prove important during the war. The RN, meanwhile, was left with no experience in aviation. This led them to place flawed requirements for new aircraft and carriers. This hampered their effectiveness when engaging land-based aircraft in the North Sea or the Mediterranean. Even so, they gave sterling and effective service throughout the war.
The French had cancelled the Normandie class of battleships, under construction at the time, following the 1922 Washington Treaty. However, they had been greatly impressed by the possibilities of the British carriers. French observers had visited HMS Argus and received copies of the plans for HMS Eagle. This led to the conversion of one of the battleships to a carrier. The ship chosen was Bearn, the least complete - and therefore the most easily converted. The conversion began in 1923, and finished in 1928. She was intended to provide reconnaissance and fighter aircraft for the battlefleet, but would later embark attack aircraft. Bearn helped the French to develop techniques and tactics for the use of aircraft carriers. However, she was heavily flawed as a carrier; she was too slow to operate with a fleet, and while she had a two-tiered hangar, only the upper level was used for active aircraft, limiting the number of aircraft carried. She saw little active service during the war, as she was too slow to operate with most of the modern French fleet. Instead, she spent her time testing newly procured aircraft and transporting aircraft purchased from the USA. After the Fall of France, she was interned and demilitarised at the port of Fort-de-France in Martinique. The French had, however, put the lessons learned from Bearn to use when designing new carriers. Two new carriers were ordered in the late 1930s, with the names usually given as Joffre and Painleve. Work began on Joffre in 1938, but progress was slow and she was far from complete when France fell in 1940. She was cancelled by the Germans, and her steel reused elsewhere.
As for the Germans, /u/kieslowskifan has previously described the troubles faced by the German carrier project; I will give a shorter discussion here. The German naval building plan for the 1940s did call for the construction of several carriers, but only one was laid down by the start of the war. This was the Graf Zeppelin. She would run into three main problems. The first was that because she was massively over-complicated. Her powerplant was one of the most modern installed in any ship at the time, using untested high-pressure turbines, with Voith Schneider vertical propellers for low speed maneuverability. Her catapult system was designed to launch her eight fighters in four minutes. These technological marvels resulted in great delays to her construction. Secondly, there was extreme interservice rivalry between the German airforce and navy; I've already discussed why this was a problem with the RN. For the Germans, things were even more serious, as the only aircraft that could be used were converted Air Force types, completely unsuitable for naval use. Finally, she ran into political problems. Hitler saw the German surface fleet as something of a failure and a distraction from his main goals. It was generally sidelined, and recieved fewer resources than other projects. Graf Zeppelin suffered from this in particular. Her fire control system was traded to the Soviet Union, her 150mm guns were needed for coastal defence sites in Norway, and shipyard space was required to repair ships damaged in the course of other operations. She was still incomplete when Hitler ordered the cancellation of construction on new capital ships following the failure of the German fleet at the Battle of the Barents Sea in December 1942.
The Italians had several carrier designs in the interwar period, and were in the process of converting several ships to carriers during the war. The Italian navy, the Regia Marina, had built a seaplane carrier, the Giuseppe Miraglia, in the early 1920s, to test naval aviation tactics. Several designs for new carriers were drawn up towards the end of the decade, under the leadership of Vice Admiral Romeo Bernotti (deputy chief of the RM and its main strategist). Benotti's designs would again fall foul of interservice rivalry. The Italian air force refused to provide aircraft for new carriers. There was also political opposition. Carriers would only be necessary for a fleet that operated outside of the Mediterranean; a fleeet that operated inside it could be supported entirely by land-based aircraft. For an Italian government whose main priorities lay inside the Mediterranean, carriers would be an unnecessary expense. Finally, there was also opposition from within the navy, though this was by no means unique to the RM. Factions within the RM would continue to raise the ideas of new carriers throughout the 1930s, to limited avail. In 1938, the Chief of the Naval Staff, Vice Admiral Domenico Cavagnari, would state in the Italian Parliament that 'a natural aircraft carrier bestriding the Mediterranean', a firm statement that no carriers would be needed for the fleet. When war began, though, these priorities were reassessed. In 1940, the liner Roma was requisitioned for conversion into the carrier Aquila (though work would not begin until 1941); in 1942, work began on converting another liner to the carrier Sparviero. Neither would be completed before the Italian surrender, though Aquila was nearly ready for trials. There was a lack of experience in the design work, as well as a lack of expertise in the shipyards; these factors, along with a lack of resources, slowed construction.
Since /u/thefourthmaninaboat has answered your actual question, I'd like to contribute a separate bit of information about the British Pacific Fleet, mostly coming from this older answer.
The British navy did actually contribute to the fighting in the Pacific in World War II -- the British Pacific Fleet, which was stood up on 22 November 1944, eventually consisted of six fleet and 15 smaller carriers, four battleships and eleven cruisers, as well as their escort and support vessels. The Royal Navy per se contributed all the capital ships, but Commonwealth nations (particularly Australia and New Zealand) contributed smaller ships and personnel.
The reason the BPF didn't exist before late in 1944 was that the British had essentially abandoned the Pacific after the sinking of the Prince of Wales and Repulse, the fall of Singapore and the Japanese raid on Ceylon in March 1942. British naval forces were concentrated on the war in Europe, and the remaining forces stationed at Trincomalee were primarily on a defensive posture to support shipping to and from India and to support efforts of the British and Commonwealth troops in the China-Burma-India theatre.
The reason the BPF was formed in late 1944 was partially due to the fact that the RN had forces come available as the Italian and German fleets had been neutralized in 1943 and 1944, and partly due to a political calculation that contributing to the victory against Japan would be important to Britain in the postwar. Particularly it was felt that British colonies captured by the Japanese should be retaken by British forces, and that Australia and New Zealand should be given help from the "mother country" to counteract the large and increasing American presence in those waters.
The American navy was not initially well-disposed to the proposal to form a British fleet for operations in the Pacific, with Roosevelt overruling his chief of naval operations Ernest King. (King was generally Anglophobic and resented the "Europe-first" strategy that had siphoned off forces from the Pacific, where he believed the Navy was critical; but he also had reasonable worries over the British ability to sustain a fleet so far from home, mastery of which had allowed for the American navy to successfully fight its way toward Japan from 1942-44.) In the event, the British did use the American fleet train for significant amounts of fuel, supplies and repair parts, though some things (particularly American aircraft that had been "Anglicized") still had to be brought from Britain.
The BPF was initially based on Sydney, but as the war front pushed northward it operated out of a forward operating base in Manus in the Admiralty Islands. The BPF was given the task of striking Japanese oil refineries, which it had proven adept at while still forming (in a joint Anglo-American raid on Surabaya in 1944) and successfully carried out attacks against refineries near Palembang in January 1945.
Operating as Task Force 57 during the invasion of Okinawa, the British fleet was given the task of suppressing kamikaze activity from the Sakashima islands and airfields in northern Sumatra; they accomplished this both by aerial and also naval attack (that is, directly shelling the airfields).
The British carriers were particularly well suited to this because of their armored flight decks -- although they were subject to heavy kamikaze attack, they were able to remain in operation through attacks that would have taken American carriers out of service for repairs (though in some cases the armored flight deck, being the strength deck for the ship, was so badly damaged that the carriers had to be written off after the war). British Seafire fighters (the navalized version of the Spitfire) were used to provide CAP for the Anglo-American fleet during the invasion of Iwo Jima, and provided excellent service.
The performance of the fleet and its aviators mean that in mid-1945, the BEF was accepted as a component force of the American fleet, rather than being shunted off to missions of its own. The BEF also participated in raids on the Japanese home islands at the close of the war, though Halsey limited these for political reasons, and had a larger role planned for Operation Downfall, the planned invasion of Japan, which was canceled after the Japanese surrender.
Most of this is drawn from David Hobbs' The British Pacific Fleet: The Royal Navy's Most Powerful Strike Force, but the USNI also has a useful web page on the history of the fleet.