1. There are NUMEROUS controlling agencies when it comes to control of aircraft and missiles aboard the ship.
2. The flight deck itself is controlled by the Airboss (the pre-requisite for wearing the yellow shirt is to be a complete Jackass). The Boss lives in a world of "Be nos" (there will be no--i.e air force regs). The Airboss is usually a Commander and will have a "mini-boss" assist him. The boss brothers sit up on the 0-10 level and oversee the entire flight deck. They definitely get the best seat in the house as your average joe sailor is down in the berthing area catching a knee buckler and can't go out and watch. Flight decks, recovery and launch ops, fueling etc are extremely hazardous to say the least. It's up to the bosses (also known as pri fly) to keep the deck safe. Usually the boss will observe aft recoveries while the mini boss will watch the launches. Should anything be observed unsafe, either of the bosses will press a suspend button to stop flight ops. After steel beach parties or your occasional shell back ceremonies, the Airbosses will coordinate FOD walkdowns on the ship. Bosses will be not so much concerned with air control so much as tactical procedures, safety, lighting, and operations are concerned.
3. The airspace itself is controlled by the TACC with supporting agencies. The Tactical Air Control Center will have several functions under its purview--missile control, aircaft control, air defense. Aircraft will check in with one of the TACC's supporting agencies--the Carrier Air Traffic Control Center--on approach or departure. They will provide RADAR vectors just like a civilian ATC facility.
4. Carrier based aircraft will squawk mode 4 and will have to do so at a pre-determined point. If an aircraft does not comply and fails to follow pre published procedures, it is going to be in a world of shit.
5. The TACC will have a sight picture from Hawkeyes, perhaps an AWACS, AEGIS, and USMC Air Defense units ashore. If an aircraft has been determined hostile (depending on the ROE), unknown, etc, the Hawkeye will provide RADAR vectors for interceptors.
6. Prior to coming close, numerous transmissions will be made over guard (121.5) to turn around and go back. Should the aircraft not comply, it will be subject to a fox 2 kill.
7. 4,5,6 are all subject to tactical conditions, of course. During admin movements (non flight ops), there is nothing per se to discourage aircraft from sightseeingvflights, etc. In fact there have been cases where former Soviet Bear Deltas have come very close for photo opportunities, etc.
8. Recovery on the deck is provided by the LSO (paddles). I've probably bored you enough already so that will do it.
Hopefully this will give you some idea.