I've more or less got my head round what DNS is all about, but there's one aspect which eludes me.

Automatic or manual settings are available in the router and the PC, but which take priority?

I'm thinking that if the router is set to auto, the DNS will be allocated by the ISP, and if the router is set to manual, then whatever I've entered will be used. If that assumption is correct, can it ever make any difference what the settings are in the PC? Would they only come into play if I was still using a USB modem?

Any enlightenment appreciated.

Afaik, the settings on your network card will override those set on the router (Obviously just for that computer).

Open up a CMD window {type CMD in the Run box (XP) or search box (Vista, 7) then press enter}.  In the Window type ipconfig /all and press enter.  The IP addresses of your DNS servers will be listed (you may need to scroll up to see this).  In my case the DNS server IP address is the address of my router, meaning that my computer looks to the router which in turn supplies the Plusnet DNS server IP addresses it obtained automatically.

You can use this test to confirm that what Jameseh wrote is correct.     

The light has dawned. 

Thanks James & RR.

The odd thing is that my computer - Vista 64 bit - doesn't as it shows the router DNS entries

 DNS Servers . . . . . . . . . . . : 212.159.13.50
                                                  208.67.220.220

Presumably it is how the router handles it as this difference has come up before

Maybe the difference is that you have told your router to use fixed DNS servers whereas my router gets them automatically from my ISP (Plusnet).  My DNS servers could change (if Plusnet changes the IP addresses of its DNS servers) so my router tells the computer, in effect, "refer to me".  Yours are fixed so get passed on to the computer.   

Good call.
Changed it to automatic - rebooted Vista and I now get the router IP