The arboretum was a botanic garden found on board several Federation starship classes, like the Constitution-class and Galaxy-class. (TOS: "And the Children Shall Lead", "Is There in Truth No Beauty?"; TNG: "Sarek", "Data's Day")
Although the arboretum was considered a recreation area where crewmembers could relax in a natural setting, it was also an area of scientific research. (TNG: "Transfigurations", "Dark Page", "Night Terrors")
Aboard the USS Enterprise-D, the arboretum was on deck 17, section 21 alpha. It contained a number of different types of plant life, including Cypirion cactus and roses. There was also a small pond. (TNG: "Genesis", "Dark Page")
Civilians were allowed to staff positions in the arboretum. Keiko O'Brien was head of the plant biology lab while she was aboard the ship. The arboretum on the Enterprise was where Keiko's husband Miles asked to marry her. (TNG: "Night Terrors"; DS9: "Paradise") Children such as Clara Sutter were also allowed to help with small projects in the arboretum. (TNG: "Imaginary Friend")
That same year, Major Kira Nerys visited the Bareil Antos's monastery on Bajor and spent time in the arboretum there, trying to repair the stone lining of a pond. It ended up rather crooked and she commented that she was sure she could destroy his entire arboretum if she stayed long enough. (DS9: "The Circle")
In 2371, Odo spent some time in an arboretum after discovering his people's homeworld. (DS9: "The Search, Part II") Later in that year and following his transfer to Deep Space 9, Miles O'Brien drew up plans to turn one of the station's cargo bays into an arboretum to cheer up Keiko, who missed being able to do botanical research. However, these plans were abandoned when Julian Bashir convinced Miles that an arboretum could only afford Keiko a mere hobbyist's level of research opportunities, when what she really craved was an extended field expedition. (DS9: "The House of Quark")
Background information Edit
The refit Enterprise seen in Star Trek: The Motion Picture appeared to have an arboretum in the lower part of its engineering hull, as trees were seen beyond the several large viewports on each side of the hull. This appears to be the original intention of designer Andrew Probert, and is confirmed in this image and many non-canon publications such as Mr. Scott's Guide to the Enterprise.
Some blueprints have identified the two large blue windows below the main shuttlebay on the Enterprise-D model as the arboretum windows, although the set had no such windows. It is possible there is more than one arboretum on the Galaxy-class.