Basically, RC means Raid Class. It is a class in Raid Battle that determines your rank in Raid Battle.
A higher Raid Class means an access to higher level missions and the difficulty of the quest increases as you progress.
Completing all normal quests for an RC unlocks the final quest which upon completion will unlock the next level of RC.
The Maximum Raid Class currently is up to Raid Class 6.
Check out more information about Raid Battles here: