An A ringer is a ringer with long experience who is qualified to teach new people how to ring.
By contrast, a C ringer is a trainee, who can only ring under the direct supervision of an A ringer.
Thanks for the explanation. Who's Who In Ornithology (Pemberton, 1997) lists many notable ornithologists (e.g. Nigel Cleere) who worked as BTO 'A' Ringer.