I have in mind the classic "rubbing of a balloon on your hair and attract small pieces of paper" experiment (I may be wrong though!).
In this case the rubber (insulator) doesn't conduct static charges because (in particular) they are static!
They do not move!
On the surface of the balloon you scratch away (rubbing on your hair) electrons leaving the remaining atoms (of rubber) positively charged!
The fact is that these charges are localized and fixed (the other side of the balloon does not attract small pieces of paper!).
When you put the charged balloon near small pieced of paper the concentration of positive charged surface Ions induces opposite "charges" in the paper: the result is the presence of a Force (attractive) between balloon and paper. The balloon doesn't move (is in your hands) while the small pieces of paper are free to get near the balloon because of this attractive Force!
Hope it is not confusing matters even more!