In keeping with the Nazi practice of extremely good recordkeeping, prisoners at concentration camps were forced to wear colored badges so they could be easily identified at a glance. Individual identification was not important, it was only a certain type of person who would need to be recognizable. To assist camp personnel with the identification, a chart was produced showing the badge colors, usage, and placement on uniforms.

Red was used for political prisoners.
Green was used for criminals.
Blue was used for emigrants.
Purple was used for Jehovah's Witnesses.
Pink was used for homosexuals, generally only males.
Black was used for lesbians and "antisocial" Germans, who were probably just unemployed.
Brown was used for "antisocial" individuals from other countries.
Yellow was used for Jewish prisoners.

Normal prisoners simply had a colored triangle on their clothing. Repeat offenders had a bar added above the triangle, in the same color. Those being punished (as though some were not) received a black dot on a white circle below the triangle. If an Aryan had broken the law by having sexual relations with a Jew, they were sent to the camps and received a star that was yellow on the bottom and had an appropriate color on top. Polish and Slavic prisoners did not have a badge of their own but were commonly classed as criminals, asocial, or political prisoners. For most prisoners, the triangle pointed down. The exception was Jewish prisoners, whose triangle pointed up. If a Jewish prisoner was also in the camp for some other reason, that prisoner would get both triangles, sewn onto the uniform such that they formed a six-pointed star. This feature was likely added into the system as a mockery of the Star of David, an important Jewish symbol.

One of these identification charts survives today, on display at the United States Holocaust Memorial Museum in Washington, D.C. (look for it near the walk-through train car). That chart is from the Dachau camp, and probably from the early 1940s. It is in the public domain and may be seen at a number of websites, including those below.

Update: I have recently learned that the black triangle wasn't really used for lesbians in the camps. Although history has recorded the black triangle for that use, the truth is that no more than a handful of lesbians were ever sent to the concentration camps. The Nazis did make their lives difficult, but they were mostly seen as redeemable because they could still bear children. The black triangle was almost exclusively used for asocial prisoners, simply because there were no lesbians to get the mark.

Sources:
http://fcit.coedu.usf.edu/holocaust/people/victims.htm
http://www.ushmm.org/

Paragraph 175
The USHMM's 2002 exhibit "Nazi Persecution of Homosexuals," online at http://www.ushmm.org/museum/exhibit/focus/homosexuals_02/
Thanks also to Millennium for additional information.

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