As we learn in the first season, the prevailing reason Sandor hates his brother stems from a confrontation the pair have as children. Gregor's mean-spirit and oppressive nature leads him to press his young brother's face to a fire after Sandor finds and takes pleasure in a toy his brother discards. The punishment continues until the family's servants intervene, but the right side of Sandor's face sustains severe burns that scar him physically and emotionally — that's why the Hound shows an intense fear of fire during the Battle for Winterfell. In addition, the books suggest that Sandor believes his brother murdered his sister and father whose death remains a mysterious hunting accident.
On the show, the brothers grow up to work together in the Kingsguard where Sandor grows increasingly disgusted by his older brother's bloodlust. This friction comes to a head during a jousting tournament to celebrate Eddard Stark's appointment as Hand to the King. Gregor loses a round to Loras Tyrell and chops the head off his own horse in frustration. He then proceeds to knock Loras off his mount and strike the man down with his sword until Sandor intervenes. The brothers engage in a bitter battle that only ends when King Robert Baratheon demands they put down their weapons.
Sandor leaves King's Landing in the second season before the brother's can resolve their life-long feud, but the men have a brief reunion at the wight demonstration in the dragon pits during the seventh season finale. The scene crackles with tension, but with the visit being one of truce-making nature, the Hound restrains himself. Instead, he takes a verbal stab at his older sibling saying, "Remember me? Yeah, you do. You're even fuckin' uglier than I am now. What did they do to you? Doesn't matter. That's not how it ends for you, brother. You know who's coming for you. You've always known."
Tough words coming from the smaller of the pair. Yet, when we last see Sandor, he's making good on his word by travelling along the road to King's Landing with Ayra Stark, who's just about the best fighting partner in Westeros.