Remember me? I sure remember you, Clarence.
– Meg, 6.10 Caged Heat
"Clarence" is a nickname for Castiel used by the demon Meg. It was first used in the episode 5.10 Abandon All Hope..., and then in the episodes 6.10 Caged Heat, 7.17 The Born-Again Identity, and 8.17 Goodbye Stranger.
The name is a reference to the movie It's A Wonderful Life, in which the angel Clarence shows Jimmy Stewart that life is worth living, and then finally gets his wings and goes to heaven.
In 9.03 I'm No Angel, Castiel adopts the name Clarence as a cover when he is human. After learning of this and recognizing it as Meg's nickname for Castiel, the Winchesters comment on the irony of Castiel using the name of a famous angel as his alias.
In 15.13 Destiny's Child, the Cosmic Entity, after adopting Meg's form to communicate, greets Castiel with Meg's trademark "Hello, Clarence" and continues to refer to him as such throughout their interactions.