MUD is a general term for text-based roleplaying game - Multi-User Dungeon has always struck me as an unhelpful definition since it applies a lot more focus on mechanical components than most MUDs actually have.
IC is in character.
I can explain this better with an example I believe. While playing as a jeweler in Minas Tirith in a MUD I had need to consult with an armorer PC to receive some in-character training on basic armorsmithing as Gondor was, at that point in the plot, in desperate need of arms and armor quickly so all of the PCs who had any sort of forge capable of metalwork were trying to make an effort of supporting the war effort. While there were lots of weaponsmiths in EST the one armorer I knew well and enjoyed playing with happened to be a Chinese player who occasionally overlapped with EST. I ended up staying up late one Saturday night and having a very fun scene with them, but there were serious logistical complications and a bunch of DMs sent back and forth to make sure we were both available.
IC is in character.
I can explain this better with an example I believe. While playing as a jeweler in Minas Tirith in a MUD I had need to consult with an armorer PC to receive some in-character training on basic armorsmithing as Gondor was, at that point in the plot, in desperate need of arms and armor quickly so all of the PCs who had any sort of forge capable of metalwork were trying to make an effort of supporting the war effort. While there were lots of weaponsmiths in EST the one armorer I knew well and enjoyed playing with happened to be a Chinese player who occasionally overlapped with EST. I ended up staying up late one Saturday night and having a very fun scene with them, but there were serious logistical complications and a bunch of DMs sent back and forth to make sure we were both available.