The Society of Shadows was a formidable and globe-spanning organization founded and helmed by Ra's al Ghul.
The Society of Shadows was founded by Ra's al Ghul, dating back to the 1800s and quite possibly even further than that. The purpose of the organization was to accomplish Ra's life goal of restoring the earth to a pure and untainted world and undo all the pollution and damage done to it by humanity, even if that meant destroying most, if not all of humanity to reach that goal.
The organization was vast and powerful, having branches and influence all over the world. It consisted of various types of people who were fiercely loyal to Ra's, including highly trained assassins, spies, scientists, and soldiers. It also had vast resources and seemingly a large system of intelligence. This is due to the fact that Ra's was able to determine Batman's true identity partly because his organization easily determined which wealthy Americans regularly amassed the kind of technology and equipment that someone like Batman would require.
The Society of Shadows is based on the League of Assassins, Ra's al Ghul's organization in the comics. It is shown that the League also exists, and works in conjunction with the Shadows. Other adaptations have adopted the "Shadows" both in name and organization.