Bishop running rings round second-rate primate (6)
I believe the answer is:
bonobo
'primate' is the definition.
(I know that bonobo is a type of Common chimpanzee)
'bishop running rings round second-rate' is the wordplay.
'bishop' becomes 'B' (chess abbreviation).
'running' becomes 'on' (eg when a machine is running it is on).
'rings' becomes 'oo' (multiple circles).
'round' means one lot of letters goes inside another.
'second rate' becomes 'b'.
'oo' enclosing 'b' is 'obo'.
'b'+'on'+'obo'='BONOBO'
(Other definitions for bonobo that I've seen before include "our nearest relative" , "Small chimpanzee" .)