What is the procedure for landing on an aircraft carrier?
Landing on "The Boat"

Glossary


Approach turn/ 180: The approach turn itself is a 180 degree turn (190 degrees at the boat because of the angled deck) that is usually broken down into 5 segments: the 180 position, the 135, the 90 (halfway through the turn), the 45 and the groove.  Each respective position indicates the number of degrees left in the turn prior to rolling out in the groove.
(The) Ball or Meatball: an orange orb of light emitted from the IFLOLS.  A green horizontal row of lights (known as the datum) indicates proper glide slope.  If the ball is below the datum, the aircraft is low, and if its above the datum, the aircraft is high.  Aircraft should strive to keep the ball on the happy side.
Ball Call: Format: "(Side-number) (aircraft type) ball, (fuel-state), (auto-throttles)" Example: "123 Rhino ball, 9.0"
Carrier Break: A type of left hand overhead performed at 800ft AGL and 350+kts (for hornet guys)
BRC: the basic recovery course.  The course at which the ship is currently traveling, which does not include the offset for the angled deck.
Groove: the final portion of the approach (what civilians might call short-final)
Initial: on BRC, 3-5nm behind the ship, 800ft AGL, 350+kts, the starting point for the day approach
Interval: the specific aircraft in the pattern whose approach will commence just prior to yours, and whose separation you are responsible for not violating.
On speed: the proper AoA that will achieve the correct hook angle to catch the target wire of the carrier.  On speed airspeed can be approximated by the aircraft weight, but the term on speed does not refer to airspeed specifically, but units of AoA.
Side number: The 3 digit number assigned to the aircraft.  This is not the buno number (military aircraft don't have N numbers), it is a squadron specific identifier.  I will hence refer to '123' as the default side number.




Carrier Procedure

For simplicity's sake we'll pick it up at initial.  Glossary items have been italicized for ease of reading.


"123 flight of x, initial", ATC will then give the required instruction.
The flight will then side-step to the right of the carrier island, and begin looking for their interval.
Reaching the carrier, the flight will then perform the carrier break on their interval, ensuring that aircraft will arrive in 60 second intervals.
Landing checklist: 3 down and locked, flaps full, hook down, antiskid off, dispenser off, on speed XXX.
Once turned downwind, the aircraft will descend from 800ft AGL to 600ft AGL (the carrier pattern is flown at 600ft), and the aircraft will ensure it has proper lateral separation (abeam distance) from the carrier so as not to overshoot or undershoot on the approach turn.
Once abeam the LSO shack (or when the white of the round down is visible), the aircraft will increase VSI to approximately 200-300fpm and begin a 27-30 degree AoB turn.  Proper time in the groove is limited to only 15-18 seconds.  Too long or too short in the groove and the LSO can wave you off, so it is paramount that the approach turn be performed at the proper position.
At the 90, altitude should be 450ft AGL, and VSI should increase to 500fpm.
At the 45 the aircraft should be crossing the wake of the carrier, altitude should be 325-375ft AGL, and the ball should appear to rise. Note: Rolling into the groove, the ball should appear either centered or slightly high.
Once in the groove, with a visible ball, the aircraft makes its ball call.  The LSO will respond with "roger ball", which is the aircraft's clearance to land.
Once established in the groove, the aircraft should be roughly 700fpm and flying the ball for glidepath.  The pilot's scan becomes: meatball, lineup, AoA, meatball, lineup, AoA, meatball, lineup, AoA (you get the point).
As the aircraft flies in the groove the runway will constantly slip to the right requiring small, almost imperceptible, lateral stick inputs (and matching power corrections to account for the loss in lift).  The pilot will strive to keep the aircraft on centerline, on speed, with a centered ball.  Note: failure to keep the aircraft on speed can result in either a bolter, or an in-flight engagement.  It is imperative the aircraft maintain the proper AoA.
Just prior to touchdown, as the aircraft passes over the ramp, the pilot's scan will shift to ball, ball, ball, ball, ball, and the touchdown should come as a surprise.  This is referred to as "flying the ball all the way to touchdown".  Pilot's that deck spot (ie, attempt to fly to a specific spot on the deck) often fly down through the glideslope at the last moment and land short.
Once on the deck, the throttles will go to full military power (just shy of AB), and when landing has been assured, then the power will be reduced and the aircraft taxied off the landing area.




Shore Procedure

Naval aviation revolves around flying at the boat.  It is imperative that the procedures used at the boat become second nature, that they become ingrained in muscle memory.  To this effect, all shore based VFR approaches will be flown in exactly the same manner as would be at the carrier, with a few minor exceptions:


Antiskid will be on
The hook (for a non-emergency) will be up
Aircraft aren't required to have 60 seconds of separation, we routinely land with aircraft taxiing down the runway.
The throttles will be retarded unless executing a touch and go.


Otherwise, naval air stations that house carrier based aircraft all have I/FLOLS installed, and the ball is flown during normal VFR approaches to the field.



Further Reading

(Edit)

The following link, provided by the Chief of Naval Air Training Command (CNATRA), is an unclassified, publicly available, training guide used to train potential Naval Aviators to land on the boat for the first time.  The procedures are nearly identical, with a few nuances related to T-45 training operations, to the procedures used daily by fleet aviators.

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Link to document provided by CNATRA.