In the Rhineland within Nazi Germany, civilians of African descent were dubbed "Rhineland bastards" (since many were fathered by occupying French-African colonial soldiers in the 1920s). Hitler had himself said in Mein Kampf:

"The Jews had brought the Negroes into the Rhineland with the clear aim of ruining the hated white race by the necessarily-resulting bastardization.”

Under Nazi rule hundreds of these "Rhineland bastards" were sterilised in the late 1930s, with the help of a special Gestapo task force. Nazi propaganda also depicted Africans as inferior and degenerate. See these posters

Outside of these measures however, there was no clear policy regarding those of African descent in the Reich. Within Germany their numbers didn't exceed 25,000, so they weren't anywhere as visible a minority as Gypsies or Jews. I'm afraid I don't know much more about the experiences of African and Asian civilians in Nazi Europe. However, I do have some knowledge about the treatment of African soldiers by Nazi forces:

The treatment was inconsistent. In fact during the war some captured French-African colonial soldiers served as volunteers in the Wehrmacht + SS, for example in the 'Free Arabian Legion'.

But some African colonial prisoners weren't so lucky. During the 1940 French campaign, Wehrmacht units massacred thousands of African prisoners. In one incident, a group of prisoners were told by their captors to run away. German tanks then proceeded to shoot and run over the Africans. White French officers who protested such brutality were sometimes also murdered by Wehrmacht troops. One of the German units known to have taken part in these atrocities is Rommel's 7th Panzer Division.

Colonial non-white prisoners who didn't join the Germans also suffered abuse in POW camps. A Vietnamese soldier who fought for France wrote about how he and other colonial prisoners, especially Africans, were treated badly by their guards (by being beaten and generally humiliated). However, unlike Soviet POWs, the majority of these prisoners survived captivity. This again highlights the inconsistency of Nazi conduct towards Africans

Sources:

Evans, Richard J. (2005). The Third Reich in Power.

Scheck, Raffael (2006). Hitler's African victims: the German Army massacres of Black French soldiers.

BLACKS DURING THE HOLOCAUST on the United States Holocaust Museum website