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Carrier vessels themselves have no main armament, and limited secondary armament. They tend to be slower than cruisers, turn poorly, and have low armour and hitpoints compared to ships at their tier level. What they do have are three types of plane squadrons that they can launch and direct anywhere on the map to engage the enemy and scout ahead.
The three types are:
Fighters - Can't attack ships but can attack other aircraft including the little scout plans some battleships and cruisers can launch. Also can be used for scouting in a pinch.
Dive Bombers - Attack craft that, as the name implies, dives in and drops a heavy bomb on a target. Hits can cause fires as well damage, but the damage tends to be lackluster.
Torpedo Bombers - Attack craft that launch a wave of torpedoes, one per plane in the squadron, in a straight line. Hits can cause a lot of damage just like ship launched torpedoes, and can cause flooding.
The plane squadrons are launched one at a time from the carrier and the player can direct the squadrons individually or grouped to any location on the map using a unique satellite view, even setting waypoints for the squadrons to lead them around enemy ships whose Anti-Aircraft firepower can kill planes in the squadrons and lower their effectiveness. A carrier has a loadout of squadrons available and may not include one of the two types; one loadout may include one of each type and another possible researchable loadout may have two fighters and two dive bombers for example. Typically higher tier carriers have more squadrons to launch and control. Its also worth noting that carriers get aircraft of their tier level so higher tiers have planes that do more damage/fight better/fly faster.
Carriers typically play back behind their own lines, trying to avoid being seen and targeted by any enemy ships. The targets for the bomber squadrons are primarily battleships and enemy carriers, although they can go after harder to hit cruisers and destroyers in a pinch. I once even drove straight at a destroyer to attack it with secondary weapons to defend a cap point once (he died).
There are significant differences between USN and IJN carrier squadrons, as well as all the stat differences which you can look up elsewhere: American carriers have 6 planes in their squadrons while Japanese carriers have 4 planes, but to compensate the latter tend to have more squadrons and have less servicing time after planes land to get more ammo and replacement planes.
The strategy for a carrier pilot in a match depends heavily on his load out, the presence of friendly carriers and their loadout, and any enemy carriers and their loadouts. A fighter heavy loadout will do more Combat Air Patrol (aka CAP) for friendly battleships and stay closer to the front lines while a bomber heavy loadout carrier can send their squadrons on a looping path around the hostile cruisers and destroyers to attack deep at the enemy rear... as long as hostile carriers are not doing serious CAP.
The end effect of carriers in the game is that all ships need to be aware not only of where the enemy ships are, but what's going on in the sky above them and what wings of death are heading their way. This makes the game extremely dynamic and at turns glorious and agonizing.