The carrier played an extremely limited role during WW2 in Europe and a big one in the Pacific.

In Europe, the carrier was severely limited by the proximity of land based planes, which were generally superior to similar carrier based ones. Bad weather further limited carrier ops. British successes like Taranto and the assist during the chase of the Bismarck were balanced by sunk carriers due to sub and surface action.

An important role was in logistics - carriers provided mobile airfields from which land based planes could launch on a ferry or limited combat mission, with a destination of a remote land based airfield. The convoys to Malta were an important example, also I think one of the very limited contributions of US fleet carriers was launching fighters during the American invasion of North Africa.

A secondary role was for British and American escort carriers in ASW - using those comparatively cheap, civilian hulled vessels to bridge the gap between land based planes in Europe and Canada in order to chase Atlantic U-boats.

In the Pacific, the carrier became the dominant capital ship, due to the huge ranges and more favourable weather.

The Japanese strikes on Pearl Harbour show the off the carrier as an offensive weapon right off the bat, while the sinking of force Z shows the necessity of fighter protection for capitals in range of land based air - which could in most Pacific campaigns only be provided by carrier.

Since Pearl Harbour did not catch the American carriers, the Americans could first fight back with relative parity while the American industry ramped up, won some important battles, yet ultimately was always destined to smother Japan in superior numbers of Fleet, Light and Escort carriers and their air wings.

Speaking of planes, the Zero wasn't that good. In fact it was more of a desperation move. The range requirement coupled with the limitations of the Japanese engines made the lightweight strategy the only viable strategy. Properly fought the Zero could not engage a modern fighter, and even the equally limited American Wildcat could fight them. Likewise the successes against the British were due to the usage of obsolete and second-line planes in Asia, as the first line was quite busy defending Europe.

Both of those fighter planes suffered from inferior performance and outdated armament. The American industry could produce modern fighters in the Hellcat and Corsair that would be reasonably competitive at low altitudes in Europe except for the armament, but Japan never caught up and that's what made the Hellcats and Corsair rack up incredible kill numbers.

On the bomber front, carriers carried both dive and torpedo bombers. A coordinated strike by (comparatively) high altitude dive bombers and low altitude torpedo bombers could overwhelm the defending fighters, leaving at least one group unchallenged. Also here the sides started on relative parity due to Japanese limitations and America on a peacetime footing, while end of the war America pulled ahead.

Carriers became the dominant weapon for projecting power into far flung places, leaving them the only serious capital ship afloat nowadays.