"'The anomaly...my ship...my crew...' I suppose you're worried about your fish, too."
- - Q to Jean-Luc Picard in 2370
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)
Background information Edit
Livingston was named for producer-director David Livingston. The fish was never called by that name in an episode, but was named behind-the-scenes by Star Trek: The Next Generation production staff. (Star Trek Encyclopedia)
According to Ronny Cox, Patrick Stewart hated Livingston's presence in the ready room, and constantly petitioned the producers to remove the fish. Stewart felt that it was inappropriate to have a captive animal in a series that valued the dignity of different species. Cox - who agreed with Stewart - 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.