Ratings, other than chief petty officers and  warrant officers, wore their branch badge on their RIGHT arm  and their rate badge on their LEFT arm. Chief petty officers wore rate badges on both cuffs [3 gilt buttons] and their branch on both collars of their jackets.   However, many CPO's did not wear branch badges and they were artificers, rightly and fairly placed above most non-artificer types because they had a higher academic status, manifest in their long apprenticeship and necessary educational qualification on joining.  Warrant officers, like artificer chief petty officers  and commissioned officers, wore rate/rank badges [respectively] only on the cuffs of their sleeves.

In the officer corps, there was a clear division between EXECUTIVE officers and officers who did not, moreover, could not, command.  Executive officer status ran from the top to the bottom of the officer corps, and any officer belonging to that status could be appointed to command a vessel or an establishment.  .  All these ranks could and did belong to the executive branch of the Navy.  The officers were specialist in warfare skills, like gunnery, torpedoes, communications, navigation, submarines, mine warfare, aviators etc, and all wore just plain gold braid on their sleeves.  Thus, they were immediately recognisable, as, though wrong to say it, seamen officers, or in cases where an officer had none of the recognised specialisations, a 'salthorse'. Aviators of course wore aircrew badges above their rings, but at a certain stage in their career in more senior ranks, they also became "seaman" officers.

The other officers were also deep specialist in their own fields, but despite their senior rank or rank seniority, they could not command.  Additionally, these officers were restricted in rank also and none transcended the rank of Rear Admiral.  The top three ranks of executive officers  had no peers with the coloured ring officers.

The following is a list of the COLOURS officers, other than medical, used to wear between their rings. Medical officers still use the colours.  The official title is "Distinction Cloth of Branch", and in the case of a commodore or a sub lieutenant, the cloth was worn beneath the single ring. 

  This is my attempt at showing you the career of a young instructor officer [schoolie] from joining to becoming the Instructor Rear Admiral, the highest rank he would have obtained.  The colour of the blue is a little different from my chart on the left, but the point is made.