Troy is introduced to us early in the first episode, Galactica Discovers Earth, Part I. While at his locker he reminisces over a picture of Serina and Apollo, and we learn that he is the character we formerly knew as Boxey all grown up. Both Apollo and Serina are mentioned to be deceased, which is not new information with regard to Serina, who died on screen (Lost Planet of the Gods, Part II), but is completely new information with regard to Apollo. His adoptive aunt Athena is not mentioned.
In the time since Apollo died, Troy has been in the care of this grandfather, Commander Adama, and has grown to become a proficient Colonial Warrior, something he promised to do way back in Saga of a Star World. As of 1980, he holds the rank of Captain.
Similar to the role of Apollo in the original series, Troy is the key figure of the series, and all of the adventures revolve around him and his colleague Dillon. Depicted as somewhat more serious than Dillon, he has a tendency to play by the book.
Troy was portrayed by Kent McCord of Adam-12 fame.
Commander Troy?
In the Battlestar Galactica novelization of The Living Legend, which covers the events of The Living Legend, Part I and The Living Legend, Part II, the introductory segment is written from the point of view of Troy, now a Commander, who was thrust into this role after the death of Adama. (This is of course, an expanded universe story based upon Galactica 1980 which is in itself not considered canonical.)
Notes
When Galactica 1980 was originally conceived, it was to have featured the characters of Apollo and Starbuck. However, neither actor was available for the project, and so instead of recasting it was decided to advance the storyline 30 years and feature two new characters, Troy and Dillon as the new leads.
Troy was a recurring character in Galactica 1980 who appeared in all but the final episode (The Return of Starbuck).
"Troy" was quoted as Boxey's given name. This fact seems to have been adopted into the Battlestar canon.