(covers information from several alternate timelines)
Cetaceans were aquatic mammals indigenous to the planet Earth and many other worlds with large oceans, such as Zadar IV. Some cetacean species were considered sentient beings. The Blue Whale was the largest known creature ever to inhabit the Earth.
In 2286, an unidentified probe tried to make contact with the then-extinct humpback whale species, and in doing so wreaked havoc on Earth's ecosystem, before Admiral Kirk managed to transport two members of the species from the 20th century back to the 23rd. The probe, apparently satisfied, reversed its course. (Star Trek IV: The Voyage Home)
In an alternate timeline created when the USS Enterprise-C disappeared from the Battle of Narendra III, Cetacean Ops was a department aboard the warship USS Enterprise-D. Shortly after the Enterprise-C emerged from the temporal rift, Joshua Kim was paged to Cetacean Ops. (TNG: "Yesterday's Enterprise")
Dolphins were present aboard the Enterprise-D in some capacity. In 2368, Geordi La Forge took the Ferengi Par Lenor to see the dolphins in order to keep him from disturbing Kriosian Ambassador Briam. (TNG: "The Perfect Mate") The dolphins may have been in the aquatics lab. (TNG: "Genesis")
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The Star Trek: The Next Generation Technical Manual gives further information about the cetaceans aboard Galaxy-class starships, stating they (a mixed group of Bottlenose dolphins and Takaya's whales) are actually crewmembers and form a guidance and navigation consultation team. These cetaceans are elite specialists in navigation, and Starfleet consults them on suggested system upgrades. The USS Enterprise NCC-1701-D Blueprints also show several large cetacean tanks within the saucer section. Although the above reference to dolphins in "The Perfect Mate" was written at the request of the manual's co-author Rick Sternbach, it was never revealed in an episode whether or not the cetaceans are crewmembers as stated here.
The idea of Dolphin crewmembers may possibly be a reference to the short series Gunbuster, which was released during 1988-89 and, during the fifth episode, briefly shows a complement of dolphins aboard the starship Excellion who work as navigators. This is supported by the fact that the technical manual lists the fictional "Takaya's whales" species as a complement of the aquatics lab, which were likely named after the protagonist of Gunbuster, Noriko Takaya. (citation needed • edit)
The early concept art for the computer game Star Trek Online includes a sketch for an aquatics lab aboard a Galaxy-class starship, which may or may not be the one mentioned in the TNG episode "Genesis".