A grandfather is a title used to describe the father of one's own parents. A great-grandfather refers to the father on one's own grandfather.
Deanna Troi's grandfather was someone she spent time with on Betazed while growing up. She later described him as a man who rarely spoke, noting that talking was for off-worlders and people who didn't know any better. When Troi was young, her grandfather would tell her stories to her telepathically as they sat by the fire. Troi would close her eyes and listen to his mind for hours on end. When Troi got older, the only time she could remember his stories was when she went home to Betazed and sat by the fireplace, adding that at times, when she was alone there, it was almost like she could hear his voice inside her head. (TNG: "Eye of the Beholder")
Riley Frazier had fond memories, while growing up in mid-24th century Texas, of picking bluebonnets with her grandfather, who carried a walking stick. (VOY: "Unity")