Two terms of art in carrier landings are "trap," which means a successful landing, and "bolter," which means the airplane touched down but the hook did not engage a wire. For an excellent and illustrated description of the event, see here.
The tail hook, however, is deceptively simple. It has to take the full weight of the airplane times two or three and transmit that load into the airplane's structure. It also has to be mounted so that it doesn't cause excessive yaw or pitching moments during the trap. It has to be resistant to bouncing off the deck on contact (which might result in it skipping over each wire, causing a bolter, but yet be able to swivel up and back to minimize the moments and loads. It can neither be too long (risking an in-flight engagement, which makes a normally hard landing even harder) nor too short (risking no engagement at all).
The Grumman F11F-1 tail hook in the picture above is one of the more unusual installations. Grumman's practice had been to locate the hook at the extreme end of the fuselage. Prior to the F11F, it had been housed within the fuselage. When it was released for landing, it slid aft and then pivoted downward. After landing, the pilot could raise it to the "stinger" position so it was clear of the wires for taxiing forward. The deck crew would then stow it after shutdown.
For simplicity and weight reduction, the F11F hook did not slide aft, but was stowed upside down with the hook point forward.
When released, it simply dropped into position. After landing the pilot would raise it to the stinger position so the Tiger could be taxied forward into parking.
For stowage, the attach point was triple-jointed so the deck crew could swing it out to the side and then rotate it upside down.
It was innovative, but it didn't catch on. For one thing, the tail hook couldn't be extended after the airplane was on the ground, precluding it from being dropped to engage the emergency field arresting gear to help abort a takeoff or to terminate a unexpectedly long landing roll.