Navy Ranks, Rates, Ratings, and Paygrades
In the enlisted branch of the Navy, a field of work or an occupation is called a rating. The ratings are abbreviated with two, sometime three letters. For example the Boatswain's Mate rating is abbreviated BM. The Cryptologic Technician (Interpretive) is abbreviated CTI.
Levels within the rating are called rates. The rate abbreviation is always added right after the rating abbreviation.
3 - Petty Officer Third Class
2 - Petty Officer Second Class
1 - Petty Officer First Class
C - Chief Petty Officer
CS - Senior Chief Petty Officer
CM - Master Chief Petty Officer
So a Boatswain’s Mate second class is written BM2. Boatswain’s Mate is the rating and second class is the rate. The BM2 is also a PO2 or Petty Officer Second Class. In speaking, we don't mix the two - you're either a Boastswain's Mate second class or a Petty Officer Second Class. Never Boatswain's Mate Petty Officer Second Class. So remember, rating is the job or occupation, while rate corresponds to the grade of the person.
While it's relatively easy to remember that a BM2 is a "Boatswain’s Mate second class," it can be confusing when someone becomes a Chief Petty Officer. It's written BMC, but a BMC is formally referred to as a Chief Boatswain's Mate or generally, Chief Petty Officer. Informally, one can say Chief, CPO, "a Boatswain's Mate Chief" or even BMC, but never Boatswain's Mate Chief Petty Officer.

This holds for Senior and Master Chiefs too. Even though it's written BMCS or BMCM, if you need to use the rating, they should be referred to as (among other things) Senior Chief Boatswain's Mate or Master Chief Boatswain's Mate. Not Boatswain's Mate Master Chief or Boatswain's Mate Master Chief Petty Officer, or, horribly wrong, Boatswain's Mate Chief, Master.
We should get it right.