Livingston was a lionfish kept by Captain Jean-Luc Picard in his ready room aboard the USS Enterprise-D for the duration of the ship's existence. The spherical aquarium was located in the corner of the ready room, next to the viewport. (TNG: "Encounter at Farpoint")
In 2364, Lieutenant Commander Data examined Livingston's fish tank very closely with a magnification glass, as the android was implementing some of Sherlock Holmes' characteristics at the time. (TNG: "Lonely Among Us")
When visiting a holographic representation of the NX-class Enterprise, Deanna Troi wondered how Captain Jonathan Archer could have survived without a fish tank in his ready room. (ENT: "These Are the Voyages...")
After being affected by Barclay's Protomorphosis Syndrome, William T. Riker attempted to break open Livingston's fish tank, possibly in order to consume him. Fortunately, he did not succeed. The fish had been affected by the Syndrome as well and was transformed into a jellyfish-like creature. (TNG: "Genesis")
Q thought Picard was probably worried about his fish as well as his ship and his crew after the captain saved Humanity from the effects of an anti-time anomaly in the Devron system in 2370. (TNG: "All Good Things...")
Livingston was given a much larger aquarium by 2371. Unfortunately, the Enterprise-D was soon destroyed, although Livingston's tank remained intact and survived the crash of the saucer section on Veridian III. (Star Trek Generations)
"Unofficial name given by the Star Trek production crew to the Australian lion fish that lived in the salt water aquarium in Picard's ready room. Livingston was named for Star Trek producer David Livingston. "I can see the resemblance," says Bob Justman." (Star Trek Encyclopedia (4th ed., vol. 1, p. 483))
The Australian lion fish was originally selected by TNG Season 1 set decorator John M. Dwyer. It later became the responsibility of Jim Mees, who coordinated with Mark's Tropical Fish of Studio City to take care of it between filming.
According to Ronny Cox, Patrick Stewart hated Livingston's presence in the ready room, and constantly petitioned the producers to remove the fish. Stewart, a strong animal rights advocate who opposes many forms of pet ownership, felt that using a captive animal in an ornamental display was "immoral" and inconsistent with TNG's core theme of exploring the inherent dignity of different species across the galaxy. Cox, who called Stewart's point "well taken", stated that Livingston's temporary removal in "Chain of Command" was thus a "sort of a bone they threw to Patrick". 
Livingston's aquarium was not readily visible in Star Trek: First Contact, but production photos of the set reveal that it was present. Livingston was not part of the set for Star Trek: Insurrection and Star Trek Nemesis.