The USS Republic (NCC-1371) was a Federation starship in service from the mid-23rd through the late 24th century.
In 2254, James T. Kirk served as an ensign aboard the Republic, along with his friend Ben Finney. During a duty shift, Finney accidentally left a circuit open to the atomic matter piles; Kirk logged the incident, and Finney was denied promotion. (TOS: "Court Martial")
In 2293, the Republic was assigned to Neutral Zone patrol during the time Operation Retrieve was proposed. (Star Trek VI: The Undiscovered Country)
By the early 24th century the Republic was assigned as a training ship for Starfleet Academy cadets. Although it would continue to serve the fleet through as late as 2374, it had not left the Sol System since the 2320s. (DS9: "Valiant")
Background
Although many fan and official lists include the Republic as a Constitution-class starship, its status is uncertain. Theories about the Republic began early on with reference materials published in the 1970s.
The ship was never shown on screen and its configuration was never canonically established. However, one early comic book from Marvel's Star Trek series depicts this as a Baton Rouge-class vessel, based on the illustration of that class by Rick Sternbach in the Spaceflight Chronology reference (and described in John M. Ford's novel "The Final Reflection"). This would make sense for the registry number and the general era the ship existed during; however, it bears no regard for the creator and producer intentions, nor is it confirmed by any filmed Star Trek, since these licensed books and materials are non-canon.
Ron Moore, the writer of "Valiant", stated that he intended the mention of the Republic to be a reference to the same ship that Kirk served on. It's debatable whether a ship that old would still be useful as a training ship over 125 years later, except possibly as a study of antiquities. However, some 21st century sea-going services continue to use sailing ships as training vessels (the U.S. Coast Guard, for example, has used the barque Eagle for cadet training since the 1940s).
At the beginning of the old DOS game Star Trek: 25th Anniversary (PC), the Enterprise engages the Constitution-class starship Republic in a short war game exercise. In the game, the commander of the Republic, Captain Richard Cornelius Patterson, sends you his congratulations if you defeat his ship in the exercise. If you lose the mock battle, he sends the message, "Better luck next time..."
Later in the game, the Enterprise finds the Republic adrift, wrecked by a duplicate version of the Enterprise, her crew slaughtered and the ship in a decaying orbit.