Verity was the wife of Sydney and the mother of John Smith in the fictional backstory John thought was his own.
Verity was a nurse. She and her husband raised John on Broadmarsh Street in Radford Parade in the district of Nottingham. (TV: Human Nature) After she and her husband died the family maid, Martha Jones, continued to serve John. (TV: The Family of Blood)
John Smith was in fact the Tenth Doctor, and these memories were a constructed part of his identity, after the Chameleon Arch made the Doctor human. (TV: Human Nature / The Family of Blood)
Behind the scenes
John's parents' names Verity and Sydney were a tribute and reference to Verity Lambert and Sydney Newman, both essential to the making of Doctor Who.