A carrier is a unit which can refuel, repair, and move air units independent of the aircraft itself. These units have three purposes. The first is a mobile air base or station. With the ability to move, carriers can strike at any point on the map, and do so repeatably, free from any one base. This purpose is primarily for Guerrilla Tactics, the classic Hit and Run. The second is flexibility. Carriers often come equipped with defenses, which are invaluable in a battle situation. The Atlantis and Czar are both experimental carriers, and are heavily armed, which helps massively when defending fragile aircraft. The third and final reason is a shield. Anti-Air units generally cannot hit Land or Sea units, and if they can, they cannot do nearly as much damage to it. This shield is useful for circumventing anti-air defenses, or simply running the Gauntlet on a mission where no one leaves alive.
Carriers are somewhat under-rated compared to their historical predecessor. While Carriers do have a niche in Supreme Commanders strategy, it is not a very big one due to the changing mechanics, and in most cases, a Carrier is helpful, but not completely necessary. A Carrier is often more useful on the larger maps were bases are spread farther apart and a mobile base that can refuel and repair on the go is invaluable.
Carriers are derived from the classic and revolutionary Aircraft Carrier, the weapon that remade World War II. World War II was different from a standard Supreme Commanders match in two ways. The first is Aircraft Range was much more important, due to the large scale of the war. To compound upon this, the second reason, was the return point for a piece of aircraft. Bombers were inaccurate at best, and for a Bomber to return it's cost in manufacture, ammunition, and personnel took many, many flights. As a result, Aircraft were only used on Suicide Missions when the need was great.
In Supreme Commanders, Aircraft are slightly different. For one, all Aircraft can make it back to base, albeit more slowly, with no fuel. This allows players to go on much longer bombing runs, and partially eliminates the need for carriers. When using Aircraft in large numbers (Triple Digits, this is Supreme Commander), a suicide run is much more profitable. In terms of strategic value, one Tier 3 Air Factory is worth anywhere from 5 to 20 Bombers, making a successful bombing run on one profitable, even if all the bombers die. When you have large numbers of bombers, you are much more likely to pass the Shield Threshold- essentially the amount of damage required to pop all the shields in a base. If one were to do the numbers, it doesn't take many T3 Strategic Bombers to destroy one T3 Air Factory.
Carriers were floating cities, equipped to support a large fighting and support staff, make repairs, and in some cases, assemble aircraft. They were not fortresses, but no Battleship could get within range without being spotted long before, and bombed down to the briny deeps.
For a list of all carriers, see the carriers category.