This page contains information regarding Star Trek: Discovery, and thus may contain spoilers.
The displacement-activated spore hub drive, commonly shortened to spore drive or s-drive, was an organic propulsion system the Federation experimented with during the 2240s and 2250s. The technology used mycelium spores harvested from prototaxites stellaviatori to jump or leap across the mycelial network. During such jumps, the ships were not in normal space but in the mycelial plane. (DIS: "Context Is for Kings", "Choose Your Pain")
Origins and concept
The spore drive was based on the ideas of two colleagues and friends, Paul Stamets and Straal, who had been working on the concept since 2244. They were eager to get to the "veins and muscles" that held the galaxy together.
Their research was based on the insight that at a quantum level, there was no difference between biology or physics, and specifically that spores were not only the progenitors of panspermia, but also the building blocks of energy across the universe. This allowed Stamets and Straal to approach physics as biology. As such, the equations involved were reminiscent of both quantum astrophysics and biochemistry. (DIS: "Context Is for Kings")
Initially confined to a lab, following the outbreak of war with the Klingons, Starfleet co-opted the research for military applications, much to the displeasure of Stamets. The two scientists were split up and given two different teams on different vessels – the USS Discovery and the USS Glenn – so they could work twice as fast. The Discovery also contained a large cultivation bay.
By late 2256, six months after the start of the war, both ships were conducting "black alert" maneuvers, in which they made jumps across the mycelial network. These jumps were associated with the ship's walls becoming damp. Containment of the spores was important to prevent accidents.
Key to the research was to increase the interval, which was expressed in Speirin, higher Speirins being associated with large displacement. Jumping was probabilistic, meaning that the longer the jump, the more possible outcomes there were. The two ships lacked the processing power to make the requisite number of calculations, and so long jumps resulted in navigational instability. Some six months after the start of the war, the Discovery had reached Speirin 12 and leaps measured in the hundreds of kilometers. This was considered a poor result by Discovery captain Gabriel Lorca. (DIS: "Context Is for Kings")
However, its sister ship Glenn made a breakthrough and achieved Speirin 240. The fact that they did not grow their own spores somehow resulted in the acquisition of "Ripper", an alien creature resembling an Earth tardigrade that lived in some kind of symbiosis with the spores and was capable of communicating with them, as well as utilizing the mycelial network. These abilities resulted in its function as a sort of "navigator" for the Glenn. Emboldened by the breakthrough, Straal reported that he was going to attempt Speirin 900, even though Stamets was concerned about the feasibility and safety of such a massive displacement. (DIS: "Context Is for Kings", "The Butcher's Knife Cares Not for the Lamb's Cry")
The Glenn managed to travel back and forth into the Beta Quadrant, a ninety light year jump, in 1.3 seconds. However, soon after, the crew of the Glenn hit a Hawking radiation firewall while exiting the mycelial plane, causing the death of its crew. (DIS: "Context Is for Kings", "The Butcher's Knife Cares Not for the Lamb's Cry")
The Glenn was later scuttled, though Ripper was beamed to Discovery and secured within that ship's own spore drive assembly. (DIS: "Context Is for Kings", "The Butcher's Knife Cares Not for the Lamb's Cry")
A demonstration of what was possible with the technology at this point in the war included putting someone in a test chamber with spores, and allowing them to see where the spores had been and were going to. The subject quickly cycled through a number of planetary destinations. (DIS: "Context Is for Kings")
After the Discovery's first successful jump using the tardigrade, Stamets transmitted the drive schematics to Starfleet. Production of the drive was begun at a classified Starfleet facility in Jefferson, Iowa on Earth. All Federation ships, starbases, and colonies were placed on alert to search for more tardigrades. (DIS: "Choose Your Pain")
It was possible to use the spore drive even while a ship was traveling at warp speed. The Discovery used this technique to return to the planet Pahvo, going from warp five to a complete stop in the process. (DIS: "Into the Forest I Go")
Following the conclusion of the war in 2257, Starfleet decommissioned the drive until a non-Human interface could be developed. In order to investigate one of the red bursts, Captain Christopher Pike authorized the use of the drive to travel 51,450 light years to the planet Terralysium, believing the crisis worthy of special dispensation. At the same time, Ensign Sylvia Tilly began working on the possibility of creating a coherent resonator to interact with the spores as a replacement, dark matter-powered navigational interface. (DIS: "New Eden")
After Discovery was reported destroyed in a battle with an armada of ships controlled by Control, Lieutenant Spock recommended to Starfleet that all remaining officers with knowledge of these events should be ordered never to speak of Discovery or its spore drive, under penalty of treason. (DIS: "Such Sweet Sorrow, Part 2") In the databases of Starfleet in the 32nd century, the Discovery was reported as destroyed as in 2258; however, there was no mention of the spore drive. (DIS: "Die Trying")
In 3189, during the retrofitting of the Discovery, a new and improved spore drive was installed. The ship's nacelles would have to be attached before the drive could be activated. (DIS: "Scavengers", "Unification III")
The reaction cube, in which the spores were injected, was a key part of the drive's operation. One possible density in the chamber was 68%. Another factor in spore drive was spore germination rate. (DIS: "Choose Your Pain", "The Wolf Inside")
|Displacement-activated spore hub drive|
|basidiosac rupture • black alert • bloom failure • excess energy cavitation • Hawking radiation firewall • mycelial network • mycelial plane • mycelial barrier • mycelium spore • prototaxites stellaviatori • reaction cube • speirin • spore chamber|
The spore drive was first identified as such on a monitor near the end of "Context Is for Kings", Lorca giving its full name in "The Butcher's Knife Cares Not for the Lamb's Cry". Prior to the latter episode, StarTrek.com referred to it as the "organic-propulsion system".
The technology was conceived during Bryan Fuller's time on the show, based on his interest in the work of Paul Stamets, the real-life mycologist Trek's Stamets was named after. (AT: "Context Is for Kings")
Some real-world scientists have criticized the story line and technology. 
Some of the drive's properties, such as its high speed, depended on a preexisting network and the need for an organic pilot, making it similar to the slipstream drive from Andromeda, a TV series based on concepts by Gene Roddenberry. The animation used in "What's Past Is Prologue" is also not unlike the one in Andromeda.
According to a fact sheet shown at a mirror universe exhibition running concurrent with San Diego Comic Con 2018, the ISS Discovery had a spore drive, which translated to approximately warp factor 1,431. 
In Star Trek Online, it is revealed that the Elachi have their own version of the Spore drive that is far superior to the one used by the Glenn and Discovery. It is possible that this enhanced spore drive is what first caught the attention of the Iconian civilization and led to their becoming a servitor species.