Mun is a term that is prevalent in the RP community (at least on tumblr–that’s the one I’m most familiar with) that refers to the writer of a given character that they play. I honestly don’t know where it came from, or when it started being used, as I had to look up what it meant when I first came across it! I am a fandom old, so I sometimes need to familiarize myself with these youngin’ terms.
Muse, on the other hand, refers to the character being written. I know you didn’t ask specifically about that, but I typically have seen them go hand in hand, so I thought it might be helpful to include that as well, just in case.