In 2285, Captain Terrell was commanding the starship USS Reliant and assigned to support the Genesis Project by searching for a suitable planet on which to test the Genesis Device. He had never met Admiral James T. Kirk, but knew of him, in part through his first officer, Pavel Chekov.
During a surface survey of what they believed to be Ceti Alpha VI, Terrell and Chekov were captured by Khan Noonien Singh and his followers, who had been exiled there some 15 years previously. (TOS: "Space Seed") Terrell and Chekov discovered that they were in fact on Ceti Alpha V, and that Ceti Alpha VI had actually exploded only six months after Khan was marooned.
In order to force Terrell to reveal their purpose on the surface, Khan infested him and Chekov with parasitic Ceti eels, which affected the cerebral cortex and allowed Khan to control them by suggestion. Using Terrell, Khan took control of the Reliant, stranded the Starfleet crew on the surface, and attacked spacelab Regula I, the location of the research for the Genesis Project. When Khan failed to find Genesis, he abandoned Terrell and Chekov on the station as a trap for Admiral Kirk.
The pair were rescued a short time later by a boarding party from the USS Enterprise, and subsequently discovered where the Genesis materials had been hidden – underground on Regula I itself. With the Genesis materials uncovered, Terrell contacted Khan with the coordinates, allowing Khan to steal the powerful Genesis torpedo.
Believing Kirk 'killed everyone they left behind' and had ordered Terrell to take possession of Genesis, David Marcus rushed Kirk in a fit of anger. Terrell fired his phaser, with Saavik tackling David. The blast instead hit Jedda, who was vaporized. Khan then ordered Terrell to kill Admiral Kirk. However, Terrell resisted this last deadly order. Under immense pain from the Ceti eel, Terrell turned his phaser on himself, choosing to commit suicide rather than murder another Starfleet officer. (Star Trek II: The Wrath of Khan)
Background information Edit
Captain Terrell was played by actor Paul Winfield. Director Nicholas Meyer cast Winfield purely because, after being impressed by the performer's acting, he wanted to work with the actor. "There was no real reason for him to be the captain of the Reliant, other than my great desire to direct him in scenes!" said Meyer. "I knew he could do it, without any question." (Star Trek: The Magazine Volume 3, Issue 5; )
In the shooting script for Star Trek II, it is revealed that Terrell and Dr. McCoy were old friends, but the dialogue that established this was cut from the final film.
The script describes him as "about 45, soft spoken and in good shape, but somewhat laconic." 
Terrell's first name is derived from the script and wasn't spoken onscreen.
Terrell is a recurring character in the Star Trek: Vanguard novel series, in which he serves as first officer of the USS Sagittarius, later becoming its captain. His mirror universe counterpart appears in The Sorrows of Empire.