Kain and Cecil were raised as sons by the King of Baron. However, both are orphans. It is mentioned that Kain's father was a great dragoon. Cecil's father...is much more complicated.
When the team reaches the top of Mount Ordeals, they meet spark of light that explains what it is to be a Paladin. After Cecil successfully completes his trial, the light refers t him as "my son". Cecil is confused and a bit disturbed.
When the party reaches the moon, they talk to FuSoYa, the sage of the Lunarians. He explains that he had a brother, KluYa who travelled to earth and married a human woman, who bore him two sons. Cecil is one of those sons, and FuSoYa says he looks a lot like his brother did at his age. Cecil, who previously thought his name sucked, realizes it could have been CeCilYa.
It is later revealed that Golbez, the villain up until this point, is actually being controlled by a greater evil. That, and that he's Cecil's brother. Cecil seems unsure how to react at this news. While he supported Kain unquestioningly, Cecil is less forgiving of Golbez. While he never outright rejects Golbez, he seems to have a hard time coming to terms with the whole "Lunarian" thing.
Golbez and FuSoYa go off to destroy Zemus in the name of the moon, but Cecil lags behind. Though he's eventually persuaded to go to the moon, it's Edge and Kain who really feel the need to stick it to Zemus. Cecil knows that going to the moon means facing another awkward family reunion.
At that meeting, Cecil acknowledges Golbez as his brother. He seems to have come to terms with it to a degree, but wheras after the game Golbez goes to live with the Lunarians, Cecil chooses to remain on earth and live out a human life. He hasn't rejected the alien side of him, persay, as he seems to have a friendly relationship with Golbez at this point. He's just embraced his humanity.