Well, here it is, another new year.

Admitting that I'm getting older with each new year, I hope I can become wiser with the passing of time.

In a nut shell:

P numbers are groupings of base metals with similar welding characteristics that have been approved for pressure boundaries when constructing pressure vessels or piping systems. Group numbers are sub-groupings within a P number that further subdivides the P number. Group numbers, as mentioned by John Boy, becomes an essential variable when notch toughness is a code requirement.

S numbers are similar to P numbers, but materials listed are not approved for pressure boundaries. They can be used in situations where there is no pressure differentials involved. I like to think of those materials as being appropriate for structural application, not pressure containing applications. Some construction codes do permit specific S numbered materials to be used in piping systems, so check your construction code.

Materials listed by ASME in P numbered and S numbered groupings are listed in AWS B2.1 under corresponding M numbered groups. Similar groupings of base metals can be found in NAVSEA welding standards as S numbers (not to be confused with ASME S numbers).

F numbers originally indicated "ease of use" for SMAW welding electrodes. The F numbers were used when qualifying welders to indicate what electrode classifications could be used for production. With the introduction of GTAW, GMAW, FCAW, SAW, etc, the F numbers don't exactly fit the "ease of use" concept, but is still considered to be an essential variable for welder qualification and procedure qualification. The F numbers used by ASME (Section IX) and AWS B2.1 have been harmonized and are the same. This wasn't always the case. Back in the "80's", B2.1 used F numbers as one means of differentiating between welding processes. It made sense to me, but the ASME faction prevailed, so that now all the ferrous filler metals used when welding with the gas shielded processes, FCAW, GMAW, OFW, etc. are grouped together as F6. A good move? Who knows? I preferred the older system used by B2.1, it clearly differentiated between the welding processes, whereas the ASME system does not.

A numbers refer to weld deposit chemistry. They only apply to ferrous metals. A number is an essential variable. In certain cases the A number listed on the PQR and WPS can be taken from the manufacturer's information. In other cases, such as when dissimilar base metals are welded, carbon steel to austenitic stainless steel, the A number should be based on a chemical analysis of the actual weld deposit. Anything that will alter the deposit chemistry, such as dilution, is an essential variable that must be addressed. 

Best regards - Al