This page contains information regarding Star Trek: Picard, and thus may contain spoilers.
Saturn, also known as Sol VI, was the sixth planet of the Sol system. It was well known for its rings (TOS: "Charlie X"; VOY: "Future's End, Part II") and was orbited by many moons; three of which were Mimas, Rhea, and Titan. (TOS: "The Changeling"; TNG: "The First Duty"; ENT: "Strange New World") It was also orbited by navigational control satellites. (TNG: "The First Duty")
An image of Saturn's rings, taken by the Voyager 2 probe in 1981, was displayed in astrometrics aboard the USS Voyager in the year 2374. (VOY: "Year of Hell", et al.) The rings of Saturn generated magnetic distortions. These distortions could make a starship invisible to another starship if the vessel could jump out of warp behind one of the planet's moons. (Star Trek)
In 2267, the position and orbital path of Saturn were depicted on "Chart 14A: The Sol System", stored in the USS Enterprise library computer. The chart was scanned by the probe Nomad in auxiliary control. (TOS: "The Changeling" production art)
On September 1, 1979, the space probe Pioneer 11 became the first Earth spacecraft to visit this planet, having traveled six and a half years and 3.2 billion kilometers. The probe discovered that this planet had a magnetic field and a magnetosphere. (TNG: "Loud As A Whisper" okudagram)
When she was a child, Rain Robinson was able to see Saturn's rings through her brother's telescope. She used to think that they looked like jewels from a pirate's treasure. All she ever wanted since then was to reach up and touch them. This was when she realized that she wanted to become an astronomer. (VOY: "Future's End, Part II")
During the early 21st century, the first manned Earth-Saturn probe was launched under the command of NASA's Colonel Shaun Geoffrey Christopher. (TOS: "Tomorrow is Yesterday") Accompanying Christopher were astronauts Fontana and O'Herlihy. (ENT: "First Flight")
An image of Saturn was contained in the library computer aboard the USS Enterprise. This data was flashed on a viewscreen when the Talosians scanned the Enterprise computer in 2254. (TOS-R: "The Cage")
By the early 24th century a shuttle route had been established "from Jupiter to Saturn and back, once a day, every day," know as "the Jovian Run." Both Edward Jellico and Geordi La Forge used to pilot shuttles on this route early in their Starfleet careers. (TNG: "Chain of Command, Part II")
|The Sol System|
|Mercury • Venus • Earth (Luna) • Mars (Deimos) • Asteroid belt • Jupiter (Io; Ganymede; Europa) • Saturn (Mimas; Rhea; Titan) • Uranus • Neptune (Triton) • Pluto|
According to the Star Trek Encyclopedia (4th ed., vol. 2, p. 254), Saturn has a "mass 95 times greater than that of Earth" and "a diameter of some 120,800 kilometers, and orbits its sun at a mean distance of 1.4 billion kilometers".
Additionally, the Star Trek Encyclopedia (4th ed., vol. 2, p. 149), Star Trek: Star Charts (p. 39, "United Federation of Planets I"), and StarTrek.com classified Saturn as a Class J gas giant.  This planet was a charter member of the United Federation of Planets in 2161.
Saturn was named after the titan (son of Uranus) from Roman mythology. It was unclear if Saturn itself was inhabited or not, though a reference made to Saturnius harem girls in TOS: "Mudd's Women" might suggest that it somehow was. The necessity of a daily shuttle run would also seem to support this possibility.
A graphic of Saturn and the orbits of four of its moons was seen in TNG: "The First Duty" (depicted to the right above). The outermost and largest moon was identified as Titan. The remaining three inner moons weren't named but it was very likely that they were Rhea, Dione, and Tethys, as they are the largest moons with smaller orbits than Titan (for instance, Iapetus was larger than both Tethys and Dione but its orbit was further away from Saturn than Titan's) and were all discovered in the 17th century and are well established.
According to Spaceflight Chronology (p. 45), the Earth-Saturn probe mission was conducted on the UNSS Lewis and Clark. Signs of past mining activity was discovered from the second and fourth moons of Saturn, first proof of the existence of extraterrestrial intelligent life. This last detail and "One Small Step" would put the mission after 2032.
Saturn was one of the planets seen in both versions of the opening sequence of Star Trek: The Next Generation beside the Earth, Luna, Sol, and Jupiter, shortly before the Enterprise-D started its exploration of the unknown space.
According to the shooting script of Star Trek: The Motion Picture, the Enterprise passed through the Saturnian system on its course to intercept V'ger. This was changed to the Jovian system for the film.