Zulu chief ran around trapping bishop bird (5,5)
I believe the answer is:
zebra finch
'bird' is the definition.
(zebra finch is a kind of bird)
'zulu chief ran around trapping bishop' is the wordplay.
'zulu' becomes 'z' (phonetic alphabet: alpha, bravo, charlie etc.).
'around' is an anagram indicator.
'trapping' means one lot of letters goes inside another.
'bishop' becomes 'b' (abbreviation used in chess).
'z'+'chief'+'ran'='zchiefran'
'zchiefran' is an anagram of 'zerafinch'.
'zerafinch' enclosing 'b' is 'ZEBRA FINCH'.
(Another definition for zebra finch that I've seen is " Small Australian bird with black and white striped face".)