Military personnel normally put their badges on their shoulders - what is that area called?
Well for the US military that have this practice (many do, I know):
Shoulder sleeve insignia or SSI as an abbreviation.
Interesting to note that when it is on the right shoulder, it indicates former service, not current.
The article linked to above notes that they are not continuing with this custom on newer uniforms, opting for a Combat Service Identification Badge (CSIB) instead. :-/
I believe that the word for shoulders is "epaulettes." Military officers (as opposed to enlisted men) are sometimes referred to as "epaulette boys," because of their shoulder badges.