Their is no such thing as A-rank or B, C and D for classifying Ranks for Shinobi. Those classifications are only used for Missions performed by the Ninja/Shinobi. The only exception (or somewhat) is S-rank:
S-rank (Sランク, Esu ranku) is one of the six classifications. It is preceded by A-rank. It is typically used when classifying techniques intended for highly experienced jōnin and Kage-level shinobi. S-rank techniques are almost always unique to a single user, and as such are trademark abilities of that user.
Missions can also be classified as S-rank, these are the highest paying and most dangerous type of mission a ninja can go on. These missions are exclusively assigned to highly skilled jōnin or large squads of ninja. S-rank missions pay at least 1,000,000 ryō.
"S-rank" is also a designation given to extremely powerful missing-nin, earning them a place in their village of origin's Bingo Book. Each member of Akatsuki is an S-rank criminal.
~Wikia