The Allies used their air power against German civilian populations through what was known as "strategic bombing." Strategic bombing is when the military attacks things like factories so as to prevent them from producing the things the enemy needs to fight. This is contrasted with more "tactical" bombing, in which the military targets the other side's military.
The Allies (the Germans did this to the British as well) were trying to destroy the German industrial capacity. Therefore, they had to bomb the cities where the factories and railyards and refineries and such were. In those days, bombing was extremely inaccurate and so many civilians ended up getting killed or made homeless by bombs that did not hit their targets.
It must be said, though, that the Allies did also simply target residential areas. They felt that it would be a good idea to do this to destroy the morale of the German civilians. As the link below says
In February 14, 1942, Bomber Command, the RAF branch in charge of strategic bombing, issued a directive that the "primary object" of the campaign was to destroy "the morale [spirit] of the enemy civilian population and in particular of industrial workers." The head of the RAF explained that the target was residential areas, "not, for instance, the dockyards or aircraft factories." The aim was to kill the people who worked in the factories, along with their families, and to destroy their homes.
So, the Allies used their air power against German civilians both accidentally (when they missed their industrial targets) and intentionally (as in the quote above).