What's the relationship between ASME and NB?

Try looking on and click on the registration tab. Would seem to give a history and answer your question there.

My motto: Learn something new every day

Also: There's usually a good reason why everyone does it that way

so, can i say, ASME is the one making the rules. NB is the one testing if the products comply with the rules??

That's more our less it. There are some nuances on that,

The design and construction of the boiler / pressure vessel is to ASME code, it is inspected and stamped by a NB certified inspector and then details are registered with the board.

My motto: Learn something new every day

Also: There's usually a good reason why everyone does it that way

I am interesting to see you use 'nuances', but i think it's true.
Here is something even more funny.
NB inspect and adult the manufacturers on behalf of ASME. If manufacturer pass it, they will have ASME certificate which is they should do, but they also will have NB certificate.
See, one visit delivers two certificates.
Funny!!!

The American Society of Mechanical Engineers writes the rules for New Construction of Boilers, Pressure Vessels and Pressure Relief Devices. The National Board of Boiler and Pressure Vessel Inspectors is made up of the Chief Inspectors of US States and Canadian Provinces (Jurisdictions) that enforce the State Boiler Codes. The National Board writes an Inspection Code (NBIC) that addresses (Part 1) Installation, (Part 2) Inspection & (Part 3) Repairs and Alterations. The ASME issues Certification to Manufacture New Code Stamped Objects. The National Board issues Certifications to repair ASME Code Stamped Objects, i.e. "R" Stamp, "NR" & "VR" Stamp Programs. The ASME Code used to call out the National Board by name as the organization that conducted shop reviews and Pressure Relief Valve Capacity Certification Testing on behalf of the ASME. However, several year back, for legal reasons, references to the National Board were removed and the term "Designated Organization" was inserted. The National Board issues an NB Stamp that may be applied to objects that meet Certification Standards and are registered with the National Board. The ASME Certifies the Quality Program of the Manufacturers of Code Stamped Pressure Relief Valves and the National Board Certifies the Capacity of the PRVs. I hope this is helpful.

JAC