Was possession of weapons prohibited, encouraged, allowed to members of some organizations, or what?
They had very strict gun laws from the Weimar government, that were made even tougher in 1938. There were the expected limitations on Jewish ownership and a requirement for "trustworthyness" (ie not opponents of the regime).
As @Nathan Cooper indicated, under Hitler, most of the civil weapons legislation was not actually Hitler's - he inherited strong gun control laws from Weimar Republic (aside from total prohibition for the Jews since 1938).
However, the laws were written in a way that allowed pretty much unlimited restrictions and limitations by people in power, within the framework of that legislation:
Note the laxensess of definitions. The last bullet point was such that pretty much anyone you didn't want to have a gun wouldn't have one.
A very detailed discussion is in the second half of the Straight Dope article here (the first half was mostly devoted to debunking an infamous Hitler's gun control "quote").