Prince Rhun was the well-intentioned but clumsy heir to the House of Rhuddlum, rulers of the Isle of Mona. He appeared in The Castle of Llyr, and returned in The High King as King Rhun. He was the son of King Rhuddlum and Queen Teleria.
Rhun was inept as a prince. Worse, he knew it. Try as he might, he always managed to do the wrong thing. Sometimes this would be simply embarrassing, such as falling into the sea as he tried to welcome the Princess Eilonwy to his ship. But at other times this proved dangerous, particularly when he went wandering off from a search party to rescue Eilonwy, fell off his horse, and got himself lost in the bargain.
That ramble may have been the best thing to happen to him, however. As Rhun, Taran, Gurgi, and Fflewddur Fflam encountered the giant Glew in a cavern below Mona, he proved to himself that he could be a good prince when he saved the others in his group from being killed by the giant. From that point forward, though he might remain feckless and impulsive, he would have the core of what a man should be.
Returning to Prydain as King of Mona, Rhun had grown to truly care about his people, even to the point of designing a new seawall for the island. Rhun being Rhun, the wall's design was flawed; as Taran pointed out, were it built, it would collapse immediately. He helped Taran, Gurgi, Coll and Prince Gwydion escape from Magg in Caer Cadarn by riding alone through the front gates and pretending that an army was coming close behind, but was killed in the process. Taran promised the dying Rhun that he would see the king's seawall properly built.