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A carrier is a type of military ship with a Hangar deck or Boat Bays. A carrier can transport and deploy smaller vessels, up to its hangar capacity. These vessels are called Parasites in Aurora.
See this Advanced Carrier Operations Tutorial.
Carriers can have any number of roles in a single ship. Carriers can be a fleet Point Defense and Fighter intercept ship. Or they can have a single design mindset such as just for carrying fighters, interceptors or drop ships for planetary assaults. You could design a jump-capable "survey mothership", with hangars to carry small survey craft. Or a recon ship could carry small, stealthy sensor ships that travel ahead of the fleet.
Carrier Fleet Operations
Carriers require a fleet for support because they are just that, a carrier. They are meant to support and add range to a fleet operations while also gaining protection from the fleet from enemy ships that close range or launch missiles.
You can build a fleet entirely based on carriers, or use them in a less offensive support role. However you do it, carriers are very important for a well rounded fleet and its operations.
A carrier also functions as a shipyard for repair purposes, but the ship to be repaired has to be docked inside the hangar. That way you can even repair armor while in space.
Note: it seems that hangars in PDCs can't repair ships in Aurora 7.1 anymore. Ship-based hangars can still repair docked ships/fighters.
Loading/Unloading Fighters onto Carrier
For carriers you build the mothership with hangars or boat bays. You can set their hangar loadout in the class design screen, but I haven't noticed that do anything yet, I may be doing it wrong. I mostly just use it to plan out how big to make the squadrons assigned to the carrier.
Once you have your carrier designed and built, you can assign fighters to it. Either go into the ship screen and assign the fighters to the carrier one at a time from the first tab, top right. Remember to push the land button if you want them to land in the hangar immediately after you assign them, that of course only works if they are in the same spot at the carrier.
Or you can do it the slightly less micromanagey way. Go to the Fighters screen (F7) and hit the "New" button at the bottom to create a new fighter squadron. They will be given a random name that you can change at the top if you wish. Then go to the reorganise squadrons tab and take ships from 'no squadron assignment' and move them into your squadron, hopefully remembering the capacity of the carrier you plan on assigning them to. After that you select the carrier you wish to assign them to, and if they are in the same location, hit 'recover' which will tell all the fighters in the squadron to land on the mothership. Then go back into the details tab to make sure their location is the same as their assigned ship. If so, you're golden. If not, something went wrong, either the squadron isn't in the same place as the mothership, or the mothership doesn't have enough capacity. Also, it's probably a good idea to keep different fighter classes in different squadrons. Fighter speed can vary a lot, making it inefficient to stick a slow fighter in with fast ones. Don't worry about having too many squadrons, a carrier can carry an unlimited number of squadrons, it's just the total number of ships that matter for capacity.
The assigning from the ships screen is quicker to do. However, making a squadron has the advantage of automatically splitting the fighters off into a squadron task group when launched, which saves a lot of time in the long run.
P.S. The downside of the squadron method is that it only works on fighters. If you want to add a larger ship to a hangar like a gunboat or a small destroyer, your only option is to do it from the ships screen.