Fat school boss is conceited person (7)
I believe the answer is:
bighead
'conceited person' is the definition.
(I've seen this before)
'fat school boss' is the wordplay.
'fat' becomes 'big' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'school boss' becomes 'head' (I've seen this before).
'big'+'head'='BIGHEAD'
'is' is the link.
Can you help me to learn more?
(Other definitions for bighead that I've seen before include "Braggart" , "Arrogant person" , "Conceited person (usually male)" , "Elder Dumbledore" , "Had gibe at informal egotist" .)