SA were the Paramilitary wing of the Nazi party. Hitler designed for them to be more a propaganda tool and to help campaign during elections with its main role essentially being to look smart and parade around and also to hang up posters and collect donations for the party, but its leadership viewed it as an actual army. This conflict eventually led to the purge of the SA.

The SS were originally bodyguards for Nazi politicians and security for party meetings. They were envisioned as the creme de la creme of the party. They eventually expanded in a variety of areas - controlling an intelligence agency, running concentration camps, a construction company, forming full-on army divisions, etc.

The Gestapo was originally a secret police agency in Prussia controlled by Prussia's minister Hermann Goering. As part of a political dispute, the Gestapo was placed under the control of Himmler and the SS and then expanded throughout all of Germany, not just Prussia.

Source: Adrian Weale - The SS: A New History