(covers information from several alternate timelines)
Saucer separation, or starship separation, was an emergency or tactical maneuver performed by some Federation starships involving the complete disconnection of the primary hull and the secondary hull.
By the 24th century, separations were designed to be more routine, with some classes being designed with reconnection abilities. Both components contained essential systems, allowing each to be independently operable for extended periods. (TNG: "Encounter at Farpoint") The Galaxy-class and Prometheus-class starships were designed to routinely separate into two or more components. (TNG: "Encounter at Farpoint"; VOY: "Message in a Bottle")
By the 32nd century, starship separation was even more routine. Techniques had been invented to allow ships to function at higher effeciency through the separation of their warp nacelles from the hull, and the 23rd-century USS Discovery was retrofitted to take advantage of this. (DIS: "Scavengers") Cleveland Booker's courier ship was capable of swiftly disassembling into numerous components during high-speed maneuvering and reassembling in various configurations, for tactical and collision avoidance purposes. (DIS: "There Is A Tide...")
On the Galaxy-class, saucer separation was primarily designed as a way for the civilian and nonessential crew complement to escape in the saucer section, while the senior staff and essential personnel engaged a threat or entered a potentially dangerous situation in the stardrive section. When a starship was required to separate, due to these circumstances, the separation command was issued by the commanding officer from the main bridge. The commanding officer, along with most of the senior staff, then transferred to the battle bridge on the stardrive section via a dedicated emergency turbolift, while a junior officer was typically left in command of the saucer module. The saucer was almost always given the directive to seek safety by retreating to a starbase or other allied territory. (TNG: "Encounter at Farpoint", "The Arsenal of Freedom")
The docking clamps connecting the ship were prepared for release, power was re-routed to primary hull sources. (TNG: "Hollow Pursuits") Evacuation of nonessential personnel from the stardrive section was conducted by security staff. (TNG: "Encounter at Farpoint") Audio warnings were given to the crew to alert them of the pending event and how much time was remaining to vehicle separation. (Star Trek Generations)
During the actual separation event, the docking latches, on the stardrive side of the ship, collapsed and retracted into the hull of the stardrive section. An auto-separation was generally used to separate the two components of the ship, although a manual option was also available. (TNG: "The Best of Both Worlds, Part II", "Encounter at Farpoint") The separation was monitored from the main bridge and battle bridge by the conn officer and operations officer. (TNG: "Encounter at Farpoint", "The Arsenal of Freedom") Staff in engineering could also provide support during a separation. (TNG: "The Best of Both Worlds, Part II")
The drive section, which contained the ship's warp drive and the majority of weapons systems, was then free to confront danger without endangering the lives of nonessential crew.
Following the crisis, the two sections could rendezvous and rejoin. Connection was generally the reverse process of separation, again with automated and manual options.
In the alternate reality, separation aboard Constitution-class starships could only be executed manually, via activation of a three-pronged control system. Once the release system was extracted from its wall panel, each lever was folded outward and rotated clockwise to an open position, after which final confirmation was required via associated control panel.
Once confirmed, an airlock door automatically closed at the ship's separation point, and the saucer would automatically separate from the secondary hull. (Star Trek Beyond)
Though highly inadvisable, it was possible to separate the saucer from the stardrive section at high-warp velocities. Used by Jean-Luc Picard during the first encounter with Q, the saucer was separated at a warp speed of 9.6. According to Data, it was highly impractical but possible, with no margin for error. The slightest mishap would have made this attempt deadly. It was also required to clear the saucer section from the stardrive section to ensure safety, because as soon as separation was over, the saucer section would start to lose speed, posing a danger to the stardrive section. However, Picard was able to successfully separate the saucer in order to protect the families of the Enterprise from the immense power of Q. (TNG: "Encounter at Farpoint")
Emergency landing of saucer section
The saucer sections of many starship classes were designed to make planetfall on a suitable planetary surface, preferably class M. The saucer was required to make a level approach, heading towards the ground at a shallow angle. The area chosen for landing had to be smooth and relatively free of obstruction. However, unless there were landing struts or similar equipment available, the vessel would generally be disabled beyond repair. (Star Trek Generations)
Examples of separation
The USS Enterprise-D used saucer separation as a way to confront the Q entity on its first mission in 2364. Captain Picard left Lieutenant Worf in charge of the saucer section. The separation was the first to be conducted at high warp.
During the reconnection of the ship, Picard ordered new first officer William Riker to conduct the docking manually. The members of the bridge crew were somewhat apprehensive about the plan, but Riker performed well. He safely rejoined the two sections without any automated assistance, proving his command ability to Picard. (TNG: "Encounter at Farpoint")
Commander Riker suggested a saucer separation shortly before the Enterprise was sent to investigate a possible Romulan attack on the freighter Batris later that year, although Picard felt it was too premature to justify such an action.
The Klingon criminals Korris and Konmel, inadvertently rescued from the Batris by the Enterprise, were fascinated by the stardrive section of the Enterprise and were excited at the prospect of hijacking it for their own use in battle. They attempted to get Worf to assist them in accessing the battle bridge and separating the ship, but fortunately, Worf was loyal to the Enterprise crew. (TNG: "Heart of Glory")
Also that year, Acting Captain Geordi La Forge separated the ship to return and fight the automated weapons drones on planet Minos. Chief Engineer Logan was ordered to take command of the saucer section and take it to Starbase 103. (TNG: "The Arsenal of Freedom")
In 2366, Commander Shelby suggested separating the ship and using the saucer to create a distraction for the Borg. Commander Riker rejected the idea, feeling the extra power from the saucer's impulse engines would be of use in the battle. Shelby briefed Picard on her plan despite Riker's objections. (TNG: "The Best of Both Worlds") Later, Riker used Picard's knowledge of the plan to his advantage during a rescue attempt to retrieve Picard, assimilated as Locutus. Following the devastating Battle of Wolf 359, the Enterprise caught up to the Borg cube, then engaged the vessel and separated. As expected, the Borg focused their attack on the stardrive section, ignoring the saucer module. This allowed the saucer to launch a shuttlecraft, masked in an antimatter spread, towards the cube. Lieutenant Commander Data and Lieutenant Worf successfully transported from the shuttlecraft to the cube and abducted Locutus, which was a crucial event leading to the defeat of the Borg in orbit of Earth. (TNG: "The Best of Both Worlds, Part II")
In 2263 of the alternate reality, the USS Enterprise suffered major damage in the Battle of Altamid, including the loss of her warp nacelles and most of her secondary hull. Due to earlier jury-rigging, the impulse engines were still attempting to draw power from the warp core so a saucer separation had to be performed if the ship was to draw power from auxiliary generators. Captain James T. Kirk chose to perform the separation himself, but was intercepted by Krall before he could finish. Having realized Kirk was in trouble, Lieutenant Nyota Uhura finished the separation herself.
Following the successful saucer separation it was determined that while they had impulse engines again, the saucer had been caught in the gravitational pull of Altamid and could not pull out. The remaining crew members abandoned ship and the saucer crashed on Altamid, damaged beyond repair. (Star Trek Beyond)
In 2267, Captain James T. Kirk considered a plan wherein chief engineer Montgomery Scott would discard the warp drive nacelles and escape with the main section when the original USS Enterprise was threatened by Vaal at Gamma Trianguli VI. Ultimately, the separation was never ordered. (TOS: "The Apple")
Kirk attempted to order Scott to disengage nacelles, and jettison if possible, in 2269, while the ship was in orbit of Excalbia, but the transmission was cut-off before the order was received. (TOS: "The Savage Curtain")
The use of a saucer separation for a Galaxy-class starship was a way to ensure crew safety.
In 2365, saucer separation was considered in order to quarantine the cargo bay of the Enterprise-D when it was feared that a plasma plague strain had escaped a containment facility. (TNG: "The Child")
When Ansata terrorists planted an implosion device on the warp core of the Enterprise-D during an attack in 2366, Captain Picard ordered preparations to be made for a saucer separation. Fortunately, Commander La Forge was able to remove the device and beam it into space before it detonated. (TNG: "The High Ground")
Also that year, the Enterprise's warp propulsion system became contaminated with invidium, causing a physical jam of the matter/antimatter injectors. The ship began to lose structural integrity as it rapidly accelerated out of control, and a saucer separation was considered by the bridge officers in order to save the crew. However, uneven warp plasma flow to the nacelles would have caused the warp field to rupture upon separation, destroying the saucer section and rendering the maneuver useless. (TNG: "Hollow Pursuits")
Saucer separation could be used as a means to prevent individuals from seizing control of a ship. Such a case occurred in 2367 when Dr. Noonian Soong used a signal to temporarily cause Data to use any means necessary in order to return to Soong's home planet. He caused an environmental system failure on the bridge, then locked himself on the bridge and severed all command functions. Data then plotted a course to Soong's planet at high warp. Picard and the senior staff, in main engineering, attempted to separate the saucer in order to restore computer control to the stardrive section. The plan was to then incapacitate the saucer with a tractor beam once the separation was complete. However, Data was able to discover the separation command and cancel it before it could be carried out. (TNG: "Brothers")
The Enterprise struck a quantum filament in 2368, disabling most of the ship's systems and causing a gradual failure in antimatter containment. Ensign Ro Laren pushed for a saucer separation to protect survivors in the saucer section, but doing so would condemn those in the stardrive section to certain death when the core breached. Deanna Troi, the senior officer on the bridge at the time, decided to delay separation and use power from the bridge to activate crucial displays in engineering. Commanders Riker and Data saw the displays right before the ship was to separate, averting the containment breach. (TNG: "Disaster")
Unfortunately, the Enterprise-D was required to separate for the same reason three years later, in 2371. Serious damage was inflicted to the ship as a result of an ambush by the Duras sisters, and the warp containment system was critically damaged causing the coolant leak. The entire ship's complement was evacuated to the saucer and it reached safety just as the drive section exploded. The ion shock wave from the blast knocked the saucer section completely disabled out of orbit and into the atmosphere of the planet Veridian III, where the ship was forced to make a crash landing. There were only light casualties; however, the Enterprise-D saucer was damaged beyond repair. (Star Trek Generations)
While on-screen only a nacelle separation was mentioned in "The Apple" and "The Savage Curtain", a saucer separation was also described as a possibility in the writer's guide for Star Trek: The Original Series, The Star Trek Guide (third revision, page 15 of the supplement). In the guide, the saucer-like section is described to be "in fact a completely self-sustaining unit which can detach itself from the galaxy drive units and operate on atomic impulse power for short range solar system exploration." 
The first visualization of a saucer separation was conceptualized by Ralph McQuarrie, as he worked on the pre-production of the abandoned 1976-1977 Star Trek: Planet of the Titans project. He stated, "I had devised a concept for the end of the film... Some alien form has designed a way to use the power of a black hole's gravity to form a spherical shroud around the black hole. If you have a dense enough material, gravity cannot penetrate it. There are two openings in the shroud that they would use to pull ships in. The saucer of the Enterprise (which was detachable) ends up in the shroud. They meet the aliens and had a dramatic finale. These two images are of the Enterprise saucer in the shroud [....] The disc of the Enterprise would separate from the rest of the ship to land on the surface of planets."  The sketches McQuarrie referred to, of the independently operating saucer section, were published in The Art of Ralph McQuarrie (pp. 124-129).
A year later, a saucer separation involving the refit Constitution-class Enterprise (NCC-1701) was storyboarded by Andrew Probert for a possible scene at the end of Star Trek: The Motion Picture. Features such as a separation line were intentionally designed into the filming model by Probert, and were good indications that the ship can separate just like its earlier form. (Star Trek: The Next Generation USS Enterprise NCC-1701-D Blueprints)
Gene Roddenberry would have preferred if, in Star Trek III: The Search for Spock, the Enterprise's saucer section had been blown up, rather than (as happens in the film) the entire ship. "If the saucer had been blown up, at the end of the picture," speculated Roddenberry, "we [could have] had a new saucer come down and reunite the two. Symbolic of the end of the story. They preferred to do it the other way." (The Making of the Trek Films, 3rd ed., p. 47)
For the refit Enterprise, Andrew Probert purposefully designed landing gear into the saucer, hidden behind the four square panels located on the saucer's concave underside. The panels were also carried over to many TMP-era designs, including the USS Reliant and USS Excelsior. Probert had intended to place landing gear on the Enterprise-D saucer as well, but became distracted by other elements and never returned to the landing gear concept. As he recalled several years later, the ship eventually paid the price for that oversight. (Star Trek: The Next Generation USS Enterprise NCC-1701-D Blueprints; Star Trek: The Magazine Volume 1, Issue 16)
The Enterprise-D was not originally intended to separate into two distinct components; the "battle section" was originally conceived as a small "D" shaped vessel which detached from the upper part of the saucer section. Probert did not learn that the producers wanted the entire saucer to separate until late in the design stage of the Enterprise-D, and had to work out how the ship would separate while keeping the already-approved overall shape of the ship. (Star Trek: The Magazine Volume 1, Issue 16)
Saucer separation was planned as a regular feature during the early days of Star Trek: The Next Generation. Several story lines, including a B-plot for "When The Bough Breaks", were to use saucer separation. (Star Trek: The Next Generation Companion (2nd ed., p. 50)) However, budget limitations at the time did not allow for extensive footage of the separated components to be shot and for the battle bridge set to be rebuilt. It was also felt that separation slowed the progress of the story. (Star Trek Encyclopedia (2nd ed., p. 431)) Unfortunately, this meant that the Enterprise was sent into a number of dangerous situations with saucer separation never being mentioned as an option.
The Enterprise-D saucer landing sequence was conceived in 1991 for the Star Trek: The Next Generation Technical Manual. Writers Ronald D. Moore, Brannon Braga, and Jeri Taylor were inspired by that information and wanted to use a crash as a cliffhanger for the sixth season of TNG, but producer Michael Piller did not like the idea. Furthermore, it would have been too costly to film on a television budget with the VFX technology available at the time. Moore and Braga later wrote the scene into Generations. (Star Trek: The Next Generation Companion (2nd ed., p. 309); The Art of Star Trek [page number? • edit]) Various storyboards for the saucer separation in that film were illustrated by Bill George, who drew the sequence with pencil during planning discussion with John Knoll, and by Ronald B. Moore. (The Art of Star Trek, p. 284; Star Trek: The Next Generation Sketchbook: The Movies, p. 18) As depicted on-screen, the sequence involved the creation of particles similar to those seen in NASA film of actual separation stagings. (Cinefex, No. 61, p. 74)
It's noteworthy that none of the other Galaxy-class starships seen in action against the Dominion or the Borg employed saucer separation. (DS9: "The Jem'Hadar", "Call to Arms", "Favor the Bold", "Sacrifice of Angels", "Tears of the Prophets", "What You Leave Behind", VOY: "Endgame")
The Excelsior-class may be capable of separation; docking latches and a battle bridge were seen on the USS Enterprise-B master systems display, and the yoke of the ship has several potential separation lines. (Star Trek Generations, production art) Similarly, the miniature of the Ambassador-class shows striping between its saucer and engineering hull similar to that of the refit Constitution-class. 
Design sketches John Eaves produced for Star Trek: First Contact seem to imply that the Sovereign-class has routine saucer separation capabilities. (Star Trek: The Magazine Volume 3, Issue 11) In addition, separation lines plus a battle bridge were evident on the master systems display on the USS Enterprise-E bridge. (Star Trek: First Contact) However, the separation abilities have not been mentioned in dialogue; in fact, the ship was referred to as being incapable of saucer separation in the first draft script of Star Trek Nemesis. Furthermore, no impulse engines are identifiable on the MSD or production blueprints. According to Mr. Scott's Guide to the Enterprise, the original Enterprise had landing gear stored underneath the two triangle-shaped hatches on the ventral side of the saucer. A third leg popped out from the cavity where the secondary hull connected to the saucer. (Star Trek: The Magazine Volume 1, Issue 16) Similar shapes were placed on the saucer of the Enterprise-E as an homage to the original ship, although designer John Eaves did not realize the fandom history behind those objects. (Star Trek: The Next Generation Sketchbook: The Movies)
The TOS novel Black Fire involved a saucer separation, after a bomb exploded on the Enterprise bridge causing severe damage to ship and crew. The ship was reconnected at a starbase facility.
The DC Comics graphic novel Debt of Honor showed James T. Kirk and T'Cel (β) separating the saucer of the Constitution-class USS Farragut when the ship comes under attack by interdimensional beasts while returning home shortly after the encounter with the dikironium cloud creature and the death of Captain Garrovick. This allowed the bulk of the crew to escape in the saucer, while T'Cel and Kirk make for an escape pod to escape the stardrive before it self-destructs.
The novelization of Star Trek Generations showed Captain Hikaru Sulu conducting an emergency drill aboard the USS Excelsior when he received news of Kirk's death from Pavel Chekov. The ship had just completed an emergency saucer separation when the communication was received.
The novel Preserver by William Shatner, Judith Reeves-Stevens and Garfield Reeves-Stevens suggested that the USS Enterprise-E was capable of separating. The TNG relaunch novel Resistance explicitly mentions the separation of the Enterprise-E as part of measures to rescue Captain Picard from the Borg during the events of the novel.
The video game Star Trek Online allows players to perform saucer separation with a number of starship classes, namely the "Exploration Cruiser Retrofit" Class (based on the Galaxy-class) and the "Multi-Vector Advanced Escort" (based on the Prometheus-class). Their Fleet variations can also do this. As well, the game-original Odyssey-class has a variation of this called "Chevron separation", which is still the same as saucer separation. Since the "Galaxy Overhaul" the Galaxy-X Dreadnought is also capable of saucer separation. The Romulan Haakona-class warbird battlecruiser, designed based on information acquired when the Romulan Star Empire stole USS Prometheus, can similarly split into two sections to fight.
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