Gerion was described as a cold man, and distant towards his three children. He traces his lineage back to Daven Lannister, the Warden of the West and cousin and brother in law of Tywin Lannister who continued the Lannister line after the death of Tywin's children. Gerion was married to Lady Rhya Lannister, and his daughter Jeyne remembers him as being heavy handed with discipline, once striking Tyrius "so hard that the welt from the ruby stone [on his ring] lasted over a month."
“Terrifying. Such that years had to pass before his children realized they no longer had to act as if the shadow of his hand were still looming over them. Or the back of his hand, rather. A good lord. Perhaps not the best father.” - Aemon Estermont
“My grandsire was fond of children. He was a jovial man. Gerion was not.” - Stafford Lannister
"Gerion wouldn’t give my brothers the time of day. He never even called them by their names. Tyrius was ‘the boy’ and your father was ‘the other boy.’ I watched the two of them clamor for his attention like nursing puppies climbing on top of each other in a desperate attempt to reach their mother’s teat.” - Jeyne Lannister to Damon Lannister
"Disciplinary." - Stafford Lannister when asked to describe Gerion
"He’d never used words when the back of his hand would work just as well at getting a point across. Gerion Lannister saved a lot of breath that way." - thoughts of Jeyne Lannister