The procedure began to be incorporated into starship classes in the 23rd century, including the Constitution-class, but once the two hulls were disconnected they could only be re-attached through external support by a starbase or shipyard. (TOS episode: "The Apple"; TOS comic: "'Til Death")
In the mid-24th century, Starfleet began to experiment with systems that were designed to allow the two sections to be able to reconnect without external support, and implemented it on several Excelsior-class vessels, including the USS Hood. As part of the training for use of the systems, all officers were required to train to perform manual docking should the need arise.
The Galaxy-class was one of the first starship classes that were designed for separation with the two hulls having essential facilities which allowed them to operate independently for periods of time. However, with the Galaxy-class the system was designed to allow civilians and non-combatants to remain aboard the saucer section, while the main crew took the secondary hull with the main armaments to meet any threat that a starship could face. (TNG episode & novelization: Encounter at Farpoint)
In 2267, the USS Enterprise was forced to separate the saucer from the engineering hull when an alien influence had forced the men and women to turn on one another, forcing Captain James T. Kirk to order the women to the saucer and the men to the engineering hull. Following the removal of the alien influence, both sections traveled to Starbase 13 for reconnection. (TOS comic: "'Til Death")
The USS Confederate performed a saucer seper during its first mission testing a new warp accelerator. When the engines began to exceed critical, without the ability to control the reaction, the crew jettisoned the saucer as a lifeboat. The stardrive hull eventually lost momentum in Romulan space, and was scuttled in a daring recovery operation by Starfleet Security. (TOS - Unlimited comic: "None But the Brave")
In 2364, Captain Jean-Luc Picard was forced to order an emergency saucer separation at warp speed following the USS Enterprise-D's first encounter with Q. While the saucer continued on to Deneb IV, the stardrive section confronted Q. The two sections were later reconnected in orbit of Deneb IV. (TNG episode & novelization: Encounter at Farpoint)
A few months later, Geordi La Forge ordered a saucer separation so that he could return to the planet Minos and rescue Captain Picard and an away team, after the Enterprise had come under attack by the Echo Papa 607 weapons system. (TNG episode: "The Arsenal of Freedom")
In early 2367, the Enterprise separated as part of a tactical maneuver to confuse the Borg and allow a small team to infiltrate the vessel and recover Captain Picard who had been assimilated by them and turned into Locutus. (TNG episode: "The Best of Both Worlds")
Two years later, Lieutenant Commander Data's judgement was impaired by nanotechnology disguised as grain that he had earlier ingested. Believing the Klingon Empire to be behind a plot to conquer the Federation, Data initiated a saucer separation, and took the stardrive section on a course towards Qo'noS. Data was stopped before he could reach Klingon space. (TNG novel: Foreign Foes)
Saucer separation is a maneuver where a Federation starship separates its saucer section from its stardrive section. In 23rd century ships, this would be used only in emergencies, whereas 24th century vessels could redock the two sections unaided. Prometheus-class ships take the concept further with their multi-vector mode mode.
The USS Swiftfire-A performed a emergency saucer separation when it was trapped in a temporal causality loop in 2374. This was done to avoid the destruction of the whole ship by a warp core breach. However due to damage to the ship's systems many of the crew in the secondary hull were not made aware of this maneuver. In the end it proved a futile effort as the saucer section was also destroyed, resetting the loop. (Star Trek: Swiftfire: "Q, Time and Again")