(covers information from several alternate timelines)
Dax was a Trill symbiont, a symbiotic lifeform that existed with the humanoid Trill as a joined species. Each joining of Dax with a host created a new, unique individual, but each individual also carried the memories of the previous hosts.
As of 2375, the Dax symbiont had been joined to nine official hosts, starting with Lela Dax in 2168 through to its latest host, Ezri. Additionally, the Dax symbiont was temporarily joined with Verad during Jadzia's first year on Deep Space 9. (DS9: "Facets", "Invasive Procedures", "Image in the Sand", "Equilibrium")
Dax was born on Trill in 2018. By 2374, it had been married six times – twice as a groom, and four times as a bride. Dax had a total of nine children – five as a mother, and four as a father. Having been a mother three times and a father twice, Dax claimed, "I wasn't very successful either way." (DS9: "You Are Cordially Invited", "Time's Orphan", "The Nagus")
Dax first served as a field docent to a Trill initiate in 2170. Over the next two hundred years, Dax became infamous for recommending the rejection of fifty-seven candidates for joining through the program. (DS9: "Playing God")
Dax held numerous positions of distinction throughout its many lives, including head of the Trill Symbiosis Commission, the leader of the Trill government, Federation ambassador, and two careers in Starfleet. (DS9: "Facets")
Of the Daxes, who were never that enthusiastic for gardening, Tobin Dax was the only one who tried gardening; however, according to Jadzia, "he had even less luck with plants than he did with women." (DS9: "The Wire")
|The hosts of Dax|
|Lela • Tobin • Emony • Audrid • Torias • Joran • Curzon • Jadzia • Ezri • Yedrin • Verad|
Although a Trill symbiont had appeared as early as TNG: "The Host", the producers wanted a new look for the Dax symbiont, the first such lifeform to be depicted on Star Trek: Deep Space Nine. "So we streamlined it," said Michael Westmore, "and made it look more like a microbe, something you'd see under a high-powered microscope." Additionally, based on the idea that the symbionts were creatures that burrowed, Westmore's team of make-up artists made the Dax symbiont pointed at one end and added some small legs. In an attempt to give the symbiont even more realism in DS9: "Invasive Procedures", Westmore inserted a vibrating pen he had found at Toys "R" Us, so the Dax symbiont could be seen moving around. (Star Trek: Deep Space Nine Companion, p. 87)
Precisely how to show the Dax symbiont (such as in "Invasive Procedures") was a concern for the production staff. "We didn't want it to look cheesy. It was always a problem showing it," explained David Livingston. "It's that whole idea of Alien; you don't really want to see too much because the audience is either going to be repulsed by it or say it's too hokey." (Captains' Logs Supplemental - The Unauthorized Guide to the New Trek Voyages, p. 63)
The British comedian Bill Bailey (who was once a team captain on the comedy panel show Space Cadets, which featured shows called "Vulcans vs. Klingons" and "Quarks vs. Roms") named his son Dax after the symbiont. He commented, "Yes, he's called Dax after a symbiont, a creature that absorbs the power of others. I may just have given him too much baggage. I'll tell him he's named after the German stock exchange." 
The symbiont's nature has saved its host's crewmates - and once even saved the universe - on at least two occasions in spin-off media. In the series Star Trek: Invasion!, where the first four crews face a race known as the Furies who once ruled the Alpha Quadrant, a temporal disruption of the wormhole results in the USS Defiant being sent back over five thousand years into the past, with the only "survivor" into the present being the Dax symbiont, kept in stasis for centuries by the alternate Julian Bashir, allowing it to pass on vital information to its past self to stop the events that lead to the destruction of the Defiant, as well as to take control of the "Unclean" - a race who absorbed others into its mass, who had been responsible for the Furies leaving the Alpha Quadrant - and destroy them. During Star Trek: Deep Space Nine - Millennium, when the crew were briefly trapped in Pah-wraith-inspired Hells to keep them occupied in the last dregs of real space/time existing after the destruction of the universe, Jadzia Dax was able to escape her Hell and free her friends in the same way because the Pah-wraiths were trying to torment all seven of her past hosts and Dax itself, causing an overload of information that allowed Jadzia to work out what was happening.