How did the Allies use their air power against the German civilian population?

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).