Red thing

It's an award called the Legion d'Honneur (the Legion of Honor) and you wear it on your lapels. It's generally awarded to deserving people who have done something exceptional.
Unfortunately, there have been some abuses and some undeserving (yet politically well-connected) people have been getting it. I think the director was trying to emphasize that these people she was trying to bring down were all politically connected members of the establishment (which they were in real life)