The Sturmabteilung (SA) were a paramilitary force created in the early days of the NSDAP as the party's muscle. Initially, they were essentially uniformed thugs used to protect party functions and leaders, and to brawl with rival factions in the running street battles that were a common feature of Weimar Germany's polarised political environment.
The SA and their leader Ernst Röhm fell out of favour in the mid-1930s, after Hitler became chancellor; as a large, rowdy and radical paramilitary force that was increasingly aligned with Gregor Strasser's left wing of the NSDAP, Hitler came to see them as a threat to his power and to his political goals (particularly as regarded the German military.)
In late June 1934, Hitler ordered the SA purged: during the Night of the Long Knives, the SA leadership were arrested; Röhm was executed. The SA continued, in much diminished form, until the decline of the Reich, but its role as any kind of power-base within the party was over.
The Schutzstaffel (SS), meanwhile, began as a much smaller and more tightly disciplined force; created in 1925 to provide the bodyguards for senior NSDAP officials (including Hitler), it expanded significantly once Heinrich Himmler took over command in 1929. The SA and SS jostled for primacy in the party hierarchy; gradually, as the SA drifted out of favour with Hitler, the SS became his primary tool for exerting control over the Party — and later, over the Reich as a whole. Ultimately, in June 1934, the SS was the weapon used to suppress the SA.
Himmler was one of the more doctrinaire National Socialists in the Nazi leadership, and viewed the mission of the SS in lofty terms: it was to become the elite vanguard and ultimate guardian of National Socialism. He turned it into a potent political power-base for himself, and an enormous shadow-state within Nazi Germany. As Reichsführer-SS, Himmler was one of the primary executors of Hitler's will, and an aggressive protector of the Nazi state from threats both internal and external.
The SS was divided into two wings: the Allgemeine-SS, which included all of the 'regular' departments of the SS, and the Waffen-SS, its military wing. Waffen-SS units were under the operational control of the military high command (OKW), but viewed themselves as 'elite' units. The Allgemeine-SS, meanwhile, was the core of the increasingly sprawling SS fiefdom.
Although the scope of the SS' responsibilities at the height of the Reich was incredibly broad, its primary function was the operation of the Nazi security apparatus. In 1939, Himmler established the Reich Main Security Office (Reichssicherheitshauptamt or RSHA), a new umbrella organisation that subsumed the state police institutions — both criminal (Kriminalpolizei or Kripo) and political (Geheime Staatspolizei or Gestapo) — into the SS. Only the regular civil police (Ordnungspolizei or Orpo) were left under the control of the Interior Ministry, the traditional administrative home of policing — and even they eventually came under the control of the SS.
The creation of the RSHA is a significant milestone because it represents the subordination of the German state's organs of control to a Party institution. The third key component of the RSHA was the Sicherheitsdienst (SD), created in 1931 as the SS' internal intelligence agency under the command of Reinhard Heydrich. The SD became effectively a sister agency to the Gestapo, gathering intelligence on enemies of the Party. RSHA commanded the Einsatzgruppen death squads that carried out ethnic cleansing operations in the rearguard of the German advance into Eastern Europe, and which were the key executors of the early stages of the Holocaust.
There were two main directorates in the SD: the Inland-SD, responsible for operations inside the Reich, and the Ausland-SD under Walter Schallenberg, responsible for foreign intelligence. In this latter function, the SD came into conflict with the military intelligence department, Abwehr, which predated the Nazi regime. The two intelligence agencies competed for primacy for years; eventually, after the Abwehr chief Admiral Wilhelm Canaris was accused of involvement in the anti-Nazi 'Schwarze Kapelle' ('Black Orchestra') network in 1944, the SD won out, absorbing the functions of the Abwehr wholesale. Canaries was executed in the aftermath of the July 1944 assassination plot against Hitler.
Outside the RSHA, the SS also developed a powerful economic base through the Economic and Administrative Main Office (Wirtschafts- und Verwaltungshauptamt), which — among other things — administered and staffed the concentration camp system, via the 'Death's Head Units' (Totenkopfverbände), and administered Nazi racial policy through the Race and Settlement Main Office (Rasse- und Siedlungshauptamt or RuSHA.)
Some good general sources on the evolution of the SS and the Nazi security state:
Some good sources on the leaders of the Nazi security apparatus: