Note about inconsiderate people who behave priggishly (6)
I believe the answer is:
prudes
'people who behave priggishly' is the definition.
'note about inconsiderate' is the wordplay.
'note' becomes 'ps' (a PS or postscript in a letter).
'about' means one lot of letters goes inside another.
'inconsiderate' becomes 'rude'.
'ps' going around 'rude' is 'PRUDES'.
(Other definitions for prudes that I've seen before include "Person excessively concerned about propriety and decorum" , "Persons of exaggerated propriety, too easily shocked" , "They are easily shocked by matters relating to sex" , "Prim or priggish people" , "Stuffy types" .)