Coriarkin is a magician who governs the island where the Dufflepuds live. As Ramandu explains toward the end of the book, Coriarkin used to be a star but he committed a crime, and Aslan punished him by putting him in charge of the Duffers. His people are pretty stupid and require a great deal of patience. He is often forced to use magic to handle them instead of explaining things to them rationally.
Coriarkin's relationship with his people is not unlike Aslan's (or, as C.S. Lewis might say, God's) with everyone else. Coriarkin explains it all to Lucy:
"One minute they talk as if I ran everything and overheard everything and was extremely dangerous. The next moment they think they can take me in by tricks that a baby would see through." (11.45)
We take this as a hint from Lewis that he thinks when people mess up their relationship with Aslan (or God), it's either because they are terrified of him or because they think they can manipulate him.
When Aslan visits him, Coriarkin asks if he will show himself to the Duffers, and Aslan replies,
"I should frighten them out of their senses. Many stars will grow old and come to take their rest in islands before your people are ripe for that." (11.7)
Coriarkin's presence on the island represents an intermediate stage for the Duffers before they can be trained to redirect their awe toward its proper object: Aslan himself.