RBI is an acronym for Run Batted In. Consequently, the plural form is still RBI, or Runs Batted In. If you hear someone say RBIs, which happens a lot, you don't necessarily have to be a jerk and correct them, but you can know that it is incorrect.
A batter earns an RBI for each run that is scored as a result of that batter's base hit, sacrifice fly, sacrifice bunt, ground out, base on balls (BB), or hit-by-pitch (HBP). If a run scores due to an defensive error or a batted ground-into-double-play (GIDP), the batter does not earn an RBI.