The carrier is where the player begins when they are introduced into the ship's bay. We know this is where your mechs are originally built. As we can see the combat mech and jet mech being built whenever you are in the game screen about to start a new timeline in-game. Other than the beginning intro where they are launched into the world we can also presume that it is the source of the breach technology utilized. As it is where Ralph Karlsson appears (to the player) when he first opens a new breach.
The carrier has two main functions, firstly building and deploying the mechs into the world. And also redeploying mechs as Lily Reed makes a statement to this in a dialogue "Last one to the carrier..." [Can't remember]. We also know this is true because the mechs don't walk to the Vek on screen, instead they fly from above landing on the ground. It is possible that the mechs may also be repaired in the carrier but this is unlikely given that we have specific missions that state repair crews from respective islands instead supply repairs to the squads. It's second and most important purpose is acting like a time travelling ship above Earth's atmosphere. As whenever we fail a timeline pilots are teleported out of their mechs back onto the ship. Where we are able to choose a pilot for the next timeline. It also explains how we can continually choose past pilots we have unlocked due to this time travel technology.
Reply to Note: It is stated as the Carrier by the pilots, also referred to as the ship/base. Carrier is probably the canonical name.