The twin princes are Lorian, Elder Prince and Lothric, Younger Prince. They can be found in the cathedral at the top of Lothric Castle. Lorian is a knight with a pale skin and long silver hair, clad in brass armor and armed with a flaming greatsword. He is apparently crippled and cannot stand on his legs. Lothric appears frail with thin pale arms and long fingers, he wears a black robe and uses healing powers to aid his brother.
Lothric and Lorian (along with Ocelotte) are the sons of King Oceiros and the queen of Lothric. The twins are said to be inseparable due to a curse. When the bell tolled to signify the fading of the flame, they rejected their duty to become Lords of Cinder and watched the fire fade from a distance. Lothric kept his sanity, unlike most Hollows, suggesting he never died.
Since Lothric was meant to become a Lord of Cinder, there is a throne dedicated to him in Firelink Shrine, but never claimed it. It's worth noting that all the other Lords were either resurrected or abandoned their thrones. Among Lothric's mentors was a skeptic who "doubted the linking of the fire", which apparently had a profound impact on the young prince's own thoughts. So, Lothric decided to ignore the Fire and let it fizzle out. He even invites the Ashen One to wait for the end along with himself and his brother.