A trophy was a physical object usually awarded to the winner of a competition or event that served as a reminder of that person's success.
In 1992, Henry Starling was awarded a trophy with certificate for excellence in astronomical sciences. He displayed this award, among others, in his office at Chronowerx Industries in 1996. (VOY: "Future's End")
In 2151, Captain Jonathan Archer asked the Suliban Silik whether his genetic engineering was given to him as payment for interfering in Klingon affairs, or was a trophy of the Temporal Cold War. (ENT: "Broken Bow")
In 2268, Doctor Leonard McCoy berated James T. Kirk for chasing after the dikironium cloud creature, which cost crewmembers their lives, saying Kirk "didn't care as long as you could hang your trophy on the wall." (TOS: "Obsession")
In 2366, on the warning of Romulan defector Alidar Jarok the USS Enterprise-D entered the Romulan Neutral Zone to investigate reports of a Romulan build up on Nelvana III but was intercepted by Commander Tomalak, who threatened to display the broken hull of the Enterprise in the capital on Romulus to both inspire the Romulan armies and serve as a reminder to any other traitors of the state. The Enterprise was able to escape thanks to Klingon reinforcements. The following year, Commander William Riker mentioned the incident to Counselor Deanna Troi after seeing Tomalak in a fake holographic simulation created by Barash, recalling the Romulan's intention to take the Enterprise home as a war trophy. (TNG: "The Defector", "Future Imperfect")
In 2370, Lieutenant Worf won Champion Standing at a Klingon Bat'leth tournament on Forcas III. He was awarded a trophy in the shape of a Klingon trifoil. The ninth place trophy was smaller consisting of a sphere fixed to a metal plate. After transferring to the space station Deep Space 9, he displayed his trophy on a shelf in the quarters that he shared with his wife Lieutenant Commander Jadzia Dax, in 2374. (TNG: "Parallels", DS9: "Resurrection")
The Kazon Ogla had trophy rooms aboard their ships where they displayed items as symbols of their victories. First Maje Razik's warship had one such room which displayed debris from a Kazon-Nistrim frigate and a tunic belonging to a dead Kazon-Relora, among others. After being captured in 2372, Commander Chakotay observed his own uniform may have made a fine trophy for the ship. (VOY: "Initiations")
The Tarkan species were said to be a powerful race who would attack any starship intruding on their space, remove the crew, settle them on a moon somewhere, and take their ship as their latest trophy. (VOY: "Darkling")
Hirogen hunters displayed the skeletons of their prey as trophies which adorned the bulkheads of their ships. In 2374, Alpha Hirogen Idrin noted that Seven of Nine's intestines would make for an interesting trophy as unusual relics were prized among the their species. Later that year, after capturing the starship USS Voyager, Alpha Hirogen Karr decorated the captain's ready room with bones and weaponary in the same manner. (VOY: "Hunters", "Prey", "The Killing Game, Part II")
In the Star Trek: The Next Generation novel Before Dishonor, "Trophy World" is the informal nickname for a Starfleet museum located on Epsilon Sigma V (β), a popular tourist attraction in the 24th century.
Star Trek Timelines features two items called "Data's trophy" and "Idrin's trophy" which can be crafted to provide stat bonuses to the player.