"Black Shirts" was an informal term for the Schutzstaffel (which apparently means "protective corps"), or SS. The shirts they wore were black.
The SS started out in 1925 as a (not "the") paramilitary wing of the German Nazi party (a.k.a. NSDAP (Nationalsozialistische Deutsche Arbeiterpartei), a.k.a. National Socialist German Workers' Party).
The SS was originally recruited from the SA (Sturmabteilung, or "Assault Divison", a.k.a. "Brown Shirts"), yet another another paramilitary Nazi group. The SS were meant to be guards for Nazi head honcho Adolf Hitler, but they grew over the years. In 1933, there were more than 50,000; when the war started, there were about 250,000. Heinrich Himmler took charge of the SS in 1929. On June 30, 1934, the SS removed the SA along with a number of other political enemies. This event is known as "the Night of the Long Knives".
Out of the SS grew the Gestapo and, eventually, the Waffen-SS, who were real combat troops. The regular SS served as police and ran the concentration camps.
The black shirt has long been popular among Nazi and Nazi-ish groups outside of Germany, such as Sir Oswald Mosley's British Union of Fascists in the 1930's and some modern neo-Nazis.
Maybe somebody with a stronger stomach than I will eventually do the research and write a real history.