What did each of the identifying badges mean?

The Nazis used triangular badges or patches to identify prisoners in the concentration camps. Different colored patches represented different groups. In Mauthausen Concentration Camp the Jews wore red and yellow triangles forming a six pointed star. The red and yellow appears to be related to the councils of the church forcing these two colors to be worn by Jews in the East and West of Europe. Red was ordered worn from the council of Buda in the thirteeenth century.

The colors and their meanings were:     

Jew	Gypsy	Bible Researcher	Homosexual
Habitual Criminal	Political Prisoner	Asocial	Emigrant

The Purple Triangle was used for those termed Bible Researcher. The group termed Jehovah's Witnesses have attempted to usurp the title. That is quite incorrect.. Jehovah's Witnesses were only part of the people making up that group. Most of the Bible Researchers (Bibelforschers) were Sabbath keeping Christians who were not part of the Orthodox Christian system dominated by the Catholics and Lutherans. Many were descended from the Albigensians and Waldensians who withstood similar persecutions by the Orthodox system during the Middle Ages. Massive numbers of these people were murdered over a long period of time. The "Asocial" category was, perhaps, the most diverse, including prostitutes, vagrants, murderers, thieves, lesbians, and those who violated laws prohibiting sexual intercourse between Aryans and Jews. In addition, while the brown triangle was used for gypsies under certain circumstances, they were more often forced to wear the black triangle categorizing them as "asocials." Some patches included letters on the triangles to further distinguish among the various groups in the camps. Most commonly, the letter indicated nationality, e.g., "F" for franzosisch (French), "P" for polnisch (Polish), "T" for tschechisch (Czech), etc., but it could also denote special sub-categories of prisoners. For example, the white letter "A" on a black triangle signified a labor disciplinary prisoner (Arbeitserziehungshaftling), while a black "S" on a green triangle identified a strafthaft, or penal prisoner. In addition, the word Blod on a black triangle marked mentally retarded inmates, and a red and white target symbol set apart those who had tried to escape. For Jewish offenders, triangles of two different colors were combined to create a six-pointed star, one triangle yellow to denote a Jew, the second triangle another color to denote the added offense. For example, a Jewish criminal would wear a yellow triangle overlayed by a green one; Jewish homosexuals wore pink triangles over yellow. Outside the camps, the occupying Nazi forces ordered Jews to wear patches or armbands marked with the star of David, though the specific characteristics of the badge (size, shape, color) varied by region. For example, some yellow stars were marked with a large "J" in the center, while elsewhere the patches had "Jude" (or "Jood," "Juif," etc.) stitched in the middle. Those who failed to wear the star were subject to arrest and deportation, a fate that frightened most Jews into compliance even though the patch subjected them to restrictions, harrassment, and isolation.

A chart of prisoner markings used in German concentration camps. USHMM

Source: Edelheit, Abraham J. and Hershel Edelheit. History of the Holocaust: A Handbook and Dictionary. Boulder, CO: Westview Press, 1994, pp. 218, 239, 266, 448. Research by Wade Cox of CCG at Mauthausen camp.