Every system whether it is peer-to-peer or domain based will have some form of authentciation and the term "network user name" actually refers to what account is required to access that particular system over the network - this is not necessarily a domain account and obviously can't be with a peer to peer network.

And so if each computer can freely access the files of each other computer and the NAS without entering any kind of password, no password need be provided when using BESR 8.5?

Peer-to-Peer networks often pass your local credentials automatically to the remote peer at an operating system level - so if the user name you log into each computer as also exists on your NAS then your will appear to see a seemless connection because the credentials are passed in the background. This background authentication operation can depend on whether it is a logged in user (to the workstation) or a service (often used by 3rd party applications such as BESR) that is making the request to access the remote system.

As such you may need to specifiy a user and password to access the NAS . If you are unsure of how to configure this on your NAS then you may need to speak to the hardware vendor (this is assuming that BESR doesn't work if you don't specify credentials and also doesn't work if you specify the same credentials that a user logs onto the workstation with.)

From computer A I can access, write, and delete files on computer B and also on the NAS via Windows Explorer. 

From computer A, I can also access, write, and delete files on computer B and also on the NAS via DOS and Windows programs.

If I install BESR on computer A and on computer B, will I need to provide a user name and password for BESR 8.5 to back up from those computers to the NAS?

I have not yet installed BESR so I am a bit in the dark as to how the installation goes.  If I do need to provide a user name and password for BESR 8.5 to back up from those computers to the NAS, where would it be provided?