The Cauldron-Born were invulnerable to all mundane weapons. A sword buried in their breast would produce only a brief halt to the deathless warrior's advance; the wound was bloodless and did no damage to the wounded Cauldron-Born. These warriors were mute, emanating a "ghostly silence", and felt no pain. In the act of their unnatural resurrection, Arawn destroyed all memory of their former selves; they could only fulfill their master's commands, and to bring others into death's cold bondage.
Their power weakened the farther and longer they stayed away from Annuvin. The Cauldon-Born were vulnerable only to the power of the magical sword Dyrnwyn, as revealed in the final volume in the series, The High King.
In the 1985 animated film, the Cauldron-Born do not exist until the Horned King -- who in this version is the main antagonist -- brings them to life with the Cauldron he has stolen from Taran and his Companions. The ghoulish sequence depicting the warriors' rebirth earned the film a PG rating.
The concept of deathless warriors brought to life by a magical cauldron originates in a story from the Mabinogion, "Branwen Daughter of Llyr". In the tale, Bran the Blessed gives the Cauldron of Life as a gift to Matholwch, King of Ireland. Through a series of events, both leaders are caught in a war, during which the Irish use the Cauldron to revive their dead. One of Bran's warriors -- Efnisien -- disguises himself as an Irish corpse, allowing himself to be cast into the Cauldron, which destroys it.
It's worth noting, however, that these "undead" warriors are never described as gaunt, silent or "hollow-eyed"; those descriptions -- and all the other unique features of the Cauldron-Born -- come from Prydain author Lloyd Alexander's fertile imagination.