When you think about it, a starship in the Star Trek universe's warp drive is nothing more than a massive antimatter bomb held barely in check by a series of force fields. As warp engines became more powerful and therefore capable of propelling larger starships with larger crews, it became apparent that a more efficient way of protecting the life of that crew was needed over the available transporters and shuttlecraft. Hence saucer separation.
Design Considerations
In its original incarnation this emergency procedure was designed to put some distance between a starship's saucer section (hopefully with the entire crew evacuated from the stardrive) and its warp core. An explosive bolt at the top of the connection plane of Constitution Class starships (like the original Enterprise) could be detonated to separate the saucer from the ship's warp drive. Since the ship's impulse engines were located on the rear of the saucer, the saucer and its crew could be moved away at a significant fraction of lightspeed. This was a one shot maneuver - reconnection could only be accomplished at a starbase.
This was as good as it got until 200 years later with the production of the Galaxy Class Starship. The Galaxy Class was a revolution, both in starship design and operational mindset. It became clear that starship crews, particularly those out of contact with their homes for extensive periods of time, functioned better with some emotional grounding. To that end, the Galaxy Class was designed with all the facilities necessary to bring the crew's families with them - classrooms, private living quarters, the whole deal. The problem with this is that, while good for operational efficiency, it put civilians directly in the line of fire should anything nasty happen.
To this end, the separation maneuver was vastly improved. No longer such a last ditch effort, the Galaxy Class was designed as two separate spaceframes that could be unlocked at will to create two independently functioning starships if necessary. The connecting surface was replete with connection hardpoints to transfer energy, turbolifts and information smoothly from one section to the other, the point being that the two ships could be redocked if possible.
In the event of an emergency (or if the ship is entering a potentially dangerous situation where destruction is likely) the two spaceframes could be separated, leaving the bulk of the crew and their families in relative safety. The electro-pneumatic docking latches are disengaged and retracted into the surface of the stardrive's connecting plane, exposing a rear-facing photon torpedo launcher on the saucer section for defense and an additional phaser array on the stardrive (or the now aptly named 'battle section') for offense. Well, MORE offense. The battle section is a TANK - fore and aft torpedo launchers, eleven independently controlled phaser arrays, a maximum sustainable speed of warp 9.96 (higher by a factor of 10 due to its greatly reduced mass) and a powerful shielding system. Yowza.
Separation and docking are generally controlled completely by the two ships' computers, though the operation can be (and has been) done manually. This is not recommended. It is also not recommended that a separation occur at high warp though this, too, has been done. Actually, they both happened in the Next Generation's first episode.
Like its ancestor, the Galaxy Class saucer section is capable of travel only at impulse. Unlike its ancestor, the ship's stardrive is equipped with its own impulse engine so that it is capable of maneuvering at sublight speeds.
The saucer was also designed to withstand a (hopefully) controlled crash-landing on a habitable planet while a rescue team was called in.
Plot Considerations
Saucer separation was envisioned as a cool visual trick by the Next Generation design team and as a way of making the Enterprise-D drastically different from Kirk's pride and joy. It was supposed to be a regular occurance. As it was, the writers discovered that using the maneuver drastically retarded the plot, slowing it so much (and eating up so much time in the process) that it was difficult to write coherent stories around the plot device. Not that that had stopped them before, still. Come the end of the Next Generation's seven season run, saucer separation was used only three times - Once in the pilot episode, once later on in the first season and once in the first episode of the fourth season.
Star Trek: Generations saw one final use of the procedure - after the enterprise was mortally wounded by an aging Klingon Bird of Prey (MAN that ticked me off) the saucer was separated to save the crew from a warp core breach and soon after crash-landed on the surface of Veridian III.