Homesickness, or feeling homesick, was an acute longing for one's home. It could be provoked by a prolonged absence, similarities between one's home and another location, or the real possibility that one would not be able to go home in the near future or ever again.
In 2267, the climate of Gamma Trianguli VI reminded Pavel Chekov of Russia, which he claimed made him homesick. (TOS: "The Apple")
During his first week at Starfleet Academy in 2350, Benjamin Sisko would beam home to New Orleans every night for dinner with his family because he was homesick. (DS9: "Explorers")
In 2366, Lieutenant Commander Data used the limited Federation knowledge of Romulus to recreate the Valley of Chula on the holodeck of the USS Enterprise-D in an unsuccessful attempt to ease the homesickness of Romulan Admiral and defector Alidar Jarok. (TNG: "The Defector")
In 2369, after Gul Dukat had lamented about missing his old office aboard Deep Space 9, Sisko sarcastically invited Dukat to drop by anytime he felt homesick. (DS9: "Emissary")
Also during that year, Lieutenant Aquiel Uhnari said in her personal logs how she was homesick for her family and Haliian traditions. (TNG: "Aquiel")
In 2371, Elim Garak explained his presence on Cardassia Prime by telling Entek that he was homesick. (DS9: "Second Skin")
In 2374, Dorian Collins became homesick while conversing with Jake Sisko about her home of Tycho City on Luna. (DS9: "Valiant")
The EMH backup module left the Kyrian and Vaskan homeworld for the Alpha Quadrant sometime after around 3074 due to a longing for home. (VOY: "Living Witness")