Dog wagging tail and circling old lady (9)
I believe the answer is:
dalmatian
'dog' is the definition.
(I know that dalmatian is a type of dog)
'wagging tail and circling old lady' is the wordplay.
'wagging' indicates an anagram (letters swing or wag around).
'circling' is an insertion indicator (circling can mean going around or containing).
'old lady' becomes 'ma' (both can mean mother).
'tail'+'and'='tailand'
'tailand' with letters rearranged gives 'daltian'.
'daltian' placed around 'ma' is 'DALMATIAN'.
(Other definitions for dalmatian that I've seen before include "one spotted" , "Spotted pet" , "Dog (from Croatia?)" , "White-coated large dog with spots" , "Old European" .)