My bike has S&S couplers (also known as S and S Machine Bicycle Torque Coupling) that allow it to be broken apart and packed into a normal-sized checked luggage box that flies at no additional charge. Getting it to fit is always a challenge, so I took photos to help me remember.
Your S&S-enabled bike will probably pack in a slightly different configuration, but you can still use this as a starting guide.
Step 1: Disassemble Bike
Take off the brakes, the rear deailleurs, the cranks, the handle bars, and fork.
Step 2: Fork and Front of Frame
Cover the frame with pipe insulation to protect the paint. Place the support bases, fork, and frame.
Step 3: Rear Wheel
Take the tires off both wheels, and place the rear in with the cassette facing down.
Step 4: Cranks
Wrap up the cranks and chain rings in a plastic bag, and pack them.
Step 5: Seat and Rear Frame
Place the seat and rear frame section.
Step 6: Front Wheel
Place the front wheel.
Step 7: Handle Bars
Fit the handle bars in.
Step 8: Tires and Finish
Squeeze the tires in. Place the top parts of the supports, add in the remains pieces, and carefully check if the box closes.
I always try the pack my bike the night before flying, to ensure I have time to do it properly.