This page contains information regarding Star Trek: Discovery, and thus may contain spoilers.
A Trill symbiont was a sentient vermiform lifeform from the planet Trill. They primarily lived symbiotically inside hosts known as the Trill, a humanoid species, also native to the planet. Through the experience gained from multiple hosts, many symbionts gained distinguished reputations from other species as Trill. They were long-lived compared to most humanoid species, and could easily live beyond 550 years.
The Trill symbionts were teardrop-shaped and could swim in their natural pools, such as the Caves of Mak'ala. The symbionts communicated with each other via electrical impulses transmitted through milky water that they inhabited. They also had a limited form of communication with certain Trill while not joined. (DS9: "Equilibrium")
A Trill symbiont joined with a new host through a surgical procedure in which it is placed in the abdomen, where it developed a connection through the narrow part of its body to the host's body. Once a symbiont was joined for over ninety-three hours, the symbiont became dependent on its Trill host and vice versa; the Trill would die within hours without a symbiont, and the symbiont could not return to the Caves of Mak'ala (if joined for the first time) nor survive a significant period of time outside of a host. (DS9: "Dax", "Invasive Procedures") Assuming that a symbiont survived a host's unexpected death away from Trill, it would be removed and stabilized for medical emergency transport back to Trill to be joined with another appropriate host. However, if the symbiont's condition deteriorated to the point that it would die en route, the only way to save its life would be joining with the closest non-joined Trill. (DS9: "Tears of the Prophets", "Shadows and Symbols"; DIS: "Forget Me Not")
After and during joining, symbionts took with them all of the memories of their previous Trill hosts so, upon joining, the new Trill host gained those memories. The brains of the symbiont and host were able to communicate, and, indeed, did not function separately. Analyzing the brain wave patterns of the symbiont separately from the host was possible, and, to a degree, the symbiont's patterns do not change significantly from host to host. (DS9: "Dax") In the case of Adira, after completing the joining with the Tal symbiont, the consciousness of their lover Gray, the symbiont's previous host, subsequently appeared to them continuously as a separate entity. In an elaborate holoprogram, Gray's consciousness was recognized by the computer as a separate entity and was able to manifest as a hologram visible to everyone inside of the program. Before the program was shut down, Gray stated that he and Tal were "stuck." (DIS: "Scavengers", "That Hope Is You, Part 2")
Although Humans could be used as a host to a symbiont, the symbiont could not remain in their bodies for long without physically harming the Human host. (TNG: "The Host") A symbiont could survive normally in a partly Trill host, however. (DS9: "Children of Time")
By the 31st century, it was possible to join Humans and symbionts together permanently. This was a practice that provoked resistance and extreme disapproval, sometimes violent, from some aspects of Trill society, as experienced by Adira Tal. (DIS: "Forget Me Not")
The Trill symbionts relied on the Trill to survive through hosts. The Symbiosis Commission oversaw the joining of the vermiform symbionts and humanoid Trills. Through the Commission, the symbionts were cared for by Trill Guardians and implanted only into Trill that had undergone a rigorous testing process, ensuring the new host had a high intelligence, balanced personality, and would not reject the symbiont. (DS9: "Dax", "Invasive Procedures", "Equilibrium")
In the 24th century, about 500 symbionts became available for joining, each year, compared to approximately 5,000 Trill becoming initiates per annum. The relatively low quantity of symbionts resulted in Trill society hiding from their general populace the fact that virtually 50% of the population was suitable for joining, and the general belief in the 24th century was consequently that only one in every 1,000 Trills was suitable for joining. This was done out of concern and fear for the symbionts, as it was felt that the sentient species could come to be viewed as mere prizes or commodities to be fought over. (DS9: "Equilibrium")
Until the mid-24th century, it was not widely known that Trill were actually a joined species, as it was considered a private matter. New hosts often had to produce false stories as to why he or she had certain memories of previous hosts. This true nature became known to the United Federation of Planets in 2367, when Ambassador Odan's male host at the time died, forcing the Commission to send a new host, Kareel, to the USS Enterprise-D for joining. Beverly Crusher became the first Starfleet doctor to learn and perform the process. (TNG: "The Host")
At some point in its long lifetime, the Sphere learned about the symbionts and their ability to merge with humanoid hosts, and knowledge of this was uploaded to USS Discovery in 2257. Shortly after traveling to the 32nd century and after learning that Adira was joined with the Tal symbiont, Captain Saru accessed the Sphere data to learn more about the species. (DIS: "People of Earth")
Rites and ceremonies
For TNG: "The Host", Michael Westmore based the design of the symbiont on a caterpillar's body attached to the head of an octopus. ("Michael Westmore's Aliens: Season Two", DS9 Season 2 DVD, special feature) A few convolutions were added to make the appliance look as if the symbiont had lots of brain matter. An air bladder was inserted into the symbiont's head and pulsated to give the impression that the symbiont was alive. The head was also colorized with a fluorescent paint, making it glow when exposed to black light during a surgery scene. (Star Trek: Aliens & Artifacts, p. 117)
For the second appearance of a symbiont, in Star Trek: Deep Space Nine pilot episode "Emissary", Michael Westmore changed the design significantly. The symbiont was "still in a similar vein shape-wise," he stated, though the revised version was smaller than its predecessor. (The Official Star Trek: Deep Space Nine Magazine issue 1, p. 28) "One [reason for changing the symbiont] was to give it more of a streamlined look [...] The way this [altered version] was designed, it was almost like a roundish type of a triangle, with these little feelers that would bend back," explained Westmore. "And by putting a lot of KY Jelly on it whenever we would do the effects, it would slide a lot easier." ("Michael Westmore's Aliens: Season Two", DS9 Season 2 DVD, special feature)
The series bible of Star Trek: Deep Space Nine stated, "The symbiont is an invertebrate, androgynous lifeform that lives within the host. It looks like a short, fat snake. Many centuries ago, the symbionts lived underground [...] [until joining with the hosts] due to an environmental disaster." 
In the first draft script of "Trials and Tribble-ations", a hypothetical evolutionary link was made between Trill symbionts and tribbles, as some xenobiologists had apparently theorized that symbionts were evolved from tribbles. Jadzia Dax consequently referred to the symbiont inside herself as "a very smart Tribble."