I assume you are asking about the yellow star Jew had to wear in public.

The provision that people persecuted by the Nazis as Jews had to wear this badge visibly sewn on their clothing was first introduced in September 1941 after Goebbels had suggested it in 1938. It served as a method of social discrimination and had the purpose to "ease" enforcement of all the anti-Jewish provisions the Nazis had introduced, e.g. Jews not being able to go to certain stores, not being allowed to ride the tram except for certain times, had a special curfew and so on.

While the Judenstern was introduced in occupied territories too over the course of the war, there were other groups who had to wear similar badges in public in the territory of the German Reich, specifically Polish and Soviet forced laborers.

Germany's war economy hinged on the fact that they had to replace all the young men drafted into the Wehrmacht with non-German citizens from other countries to work in factories and in the agrarian sector. The Germans knew this for they started to experience a distinct labor shortage already before the war. And so, in 1939 following the invasion of Poland, the decision was made by the Reich leadership to massively and forcefully draft Poles and other citizens of occupied territories as workers into the German war effort. It is estimated that about 12 million people were drafted as forced laborers during the war and at the height of the forced labor system, in August 1944, about 7 million people or 25% of the entire work force in the Reich were foreign forced laborers.

The decisions to do so lead to internal discussions in the Nazi regime because from an ideological standpoint people like Heinrich Himmler were not all too happy about bringing millions of allegedly "inferior" people to Germany and let them mingle with the German populace. Hence, the SS and Himmler insisted to introduce severe restrictions against the foreign forced laborers. The so-called Poles Decrees in 1940 and Soviet Decrees in 1941 imposed a whole host of discriminatory measures against foreign forced laborers. They too were forbidden to use public transport, visit German restaurants, ride bicycles, shop in certain places, use public telephones and so on and so forth.

Included in these provisions was a similar measure that would ease enforcement of them and contribute to a public stigmatization of these groups, namely that they were forced to wear badges on their clothing in public. Poles had to wear the Polish badge, a capitalized purple P on yellow, while forced laborers from the Soviet Union had to wear the ±Ostabrieterabzeichen* (Eastern Workers' badge), the capitaliized word "East" in white on blue. Here e.g. is the picture of a woman wearing this badge while working in an IG Farben factory.

They were aside from Jews the only groups who had to wear such badges in public while not concentration camp prisoners.

Concentration camp prisoners had to wear badges too. The badge system for CC prisoners has its origins in the early concentration camp system when prisoners who were arrested for political reasons had to wear some kind of red badge. It was most likely inspired by both the Weimar prison system and the Reichswehr military punishment camp system as well as POW camps in WWI. The first use of colored triangles dates back to Dachau under Theodor Eicke. In 1938 it was standardized by Himmler and the Inspectorate of the Concentration Camps. This is where this famous chart comes from.

Who spearheaded the design and chose the colors, we don't know because the sources were most likely destroyed before the end of the war. Also, and similar to the nominal "rules" that existed in the camps, we do not know how closely this chart and its classification system were followed in subsequent years. It's possible to say with absolute certainty that red, green, and black triangles were used as well as yellow Stars of David. We also have uniforms that bear some kind of large marking on the back denoting the inmates who wore them as likely to escape or belonging to a Sonderkommando.

It is however important to mention that the famous chart linked above implied some kind of specific order that was never that well implemented. The chart is from the Dachau camp from the 1930s and not only does the English translation contain false info (blue triangles, the "emigrants", were not for forced labor but historically almost exclusively used for Germans who had fought on the republican side in the Spanish Civil War during their imprisonment in Mauthausen) but like the system of tattoos in Auschwitz at first suggest a strict system when historically, there really wasn't.

The problem is that for some of these symbols, no evidence has been found that they were really used. While we have either photographs, testimony or artifacts that show that these colors were used as well as the double triangle for Jews, for the race defilers symbols, the brown armband, the armed forces triangle and the repeater symbols there is an absence for evidence that they were ever used within the CC system on a broad scale. Similarly, there are reports that escape subjects, instead of getting the circle with the red dot, had instead a target painted on the back of their uniform in some camps.

The problem is that while maybe this chart was strictly adhered to in the 30s and maybe shortly after the war started, by the point when the Concentration Camp system expanded massively, first in 1938, then in 1940 and finally in 1942 swelling to over 500.000 inmates from what evidence we have, this system was not strictly adhered to beyond giving somebody the batch for which they were arrested and the extra badge for Jews.

As with all official regulations for the camp system, like the rules that guards were technically not allowed to hit prisoners (obviously that wasn't adhered to), we need to differentiate between the rules in place and practices of the reality of the camp system. In the reality of the system, it was mostly the basic triangles used with the additional one for Jews. All the rest like the repeaters and so on, are hardly documented and it is very likely that if they saw use at all, it was on a very small scale and maybe only in a limited number of camps.

So, in summary, Jews and forced laborers had to wear badges in public on their civilian clothing from 1940 and 1941 respectively forward. Concentration Camp prisoners also had to wear badges on their uniform and these followed a differentiated system, however the famous chart that always pops up in these discussions is not the best source to understand how they were used in practice. For the latter Lizou Fenyvesi's article Reading Prisoner Uniforms: The Concentration Camp Prisoner Uniform as a Primary Source for Historical Research is interesting further reading.