From the website:
The Feldgendarmes were the uniformed military police of Germany. Under the Nazi regime they were under the German Army High Command. They received full infantry training and were given extensive police powers. Waffen-SS Feldgendarme formations also existed.
Their main role during the early part of the war was policing territory under German occupation – anything from traffic control through to hunting partisan resistance fighters. However, as the tide of war turned against the Third Reich, the Feldgendarmes were used to enforced discipline within the Wehrmacht, searching for malingerers and executing deserters.