Terlina III was the uninhabited third planet in the Terlina system.
In 2336, following the destruction of the colony at Omicron Theta, Dr. Noonian Soong and his wife Juliana O'Donnell fled to the jungles of Terlina III and continued their work on androids. Juliana (who was critically injured in the escape and was replaced by an android), bore her memories. She eventually wearied of the solitude and left the planet. (TNG: "Inheritance")
In 2367, Soong summoned Data to Terlina III so that he could install the new emotion chip that he had developed. Unfortunately, his signal also summoned Lore, who stole the chip and killed Soong. (TNG: "Brothers")
According to the script, the pronunciation for Terlina is "tur-LEE-nah". [1]
"Brothers" did not identify Soong's planet by name; it was not called Terlina III until "Inheritance".