Okay step back people, I got this.
The Gestapo were not, technically, a Nazi organization, because they were an official part of the the German government and incorporated several pre-Nazi organs into a unified part of the national police. The name is a diminutive; Geheime Staatspolizei is the full name, or Secret State Police. Their purpose and portfolio was to keep watch against and suppress political subversion and domestic radicalism, and on the surface wasn't much more threatening than the way the FBI keeps half-interested track of domestic radical groups with the potential to go terrorist.
Of course the Gestapo got a lot more manpower and a lot more ability to be free with its surveillance and questioning and arrests and things went downhill. This wasn't helped by the fact that the Gestapo reported to the man who was the head of the SS, and eventually, was integrated into a unified police administration including the Kripo (regular cops), Gestapo, and the SS Sicherheitsdienst (which did foreign and domestic intelligence gathering, counterintelligence, and several other functions related to security).
The SS originated as a bodyguard service for senior Nazi Party members and part of the brown-shirted SA, hence the name Schutzstaffel, or Protection Squadron, and its paramilitary stylings. It rapidly outgrew this role, casting itself as the elite of the Nazi Party. It had numerous branches and offshoots. This include, as others have alluded to, the Waffen SS combat troops, but also the Sicherheitsdienst or SD mentioned above, running the concentration camps, death squads which followed the army around for on-the-spot execution of people the Nazis didn't like, a separate court system and police which was the only group able to try an SS member for a crime, and finally personal guard groups for Hitler and his residences. (And that's just the parts that were considered main SS; there were several civilian auxiliaries including one for women, who couldn't become true members of the SS.)
In essence the SS was a mini-country unto itself within Germany.
The two organizations were not the same; the Gestapo essentially succumbed to SS control but it was never a formalized part of the SS organization nor was membership in the Nazi party or the SS required to be a member of the Gestapo.
