I've of the opinion that the letters don't actually STAND for anything, as is the case with postal codes (the Canadian equivalent of zip codes, for those who don't know).
Now Star Trek hasn't been 100% consistent on this, but my assumption is this:
NCC = Operational Starfleet vessel.
NX = Experimental Starfleet vessel prototype not yet cleared for active service. Homage to the NX-class ships that served in the mid-22nd Century.
NAR = Non-Starfleet Federation vessel. (Unclear if this is used government vessels, civilian vessels, or both). It may only apply to vessels operated by humans
, since non-Starfleet Vulcan ships like the T'Pau had a registry that began with NSP. (Perhaps there's a different registry for vessels operated by each member race, i.e. Andorians, Tellarites, Betazoids, etc.)
Not sure we'll ever get a sensible explanation of what the USS is. Yes, I know they may have said "United Space Ship" or "United Star Ship" at some point, but what exactly is united
about it? Like the concept of no money and of synthehol, it may be best not to look at that too closely.