A child born to an American parent and a German parent acquires both American and German citizenship at birth, regardless of place of birth. Neither country requires a person born under these circumstances to choose between American and German citizenship. They may keep both for life.
While Germany recognizes the concept of dual nationality, for most purposes it considers a dual national in Germany a German citizen only. Thus, the ability of the U.S. Embassy and consulates to provide assistance to an American-German dual national in Germany may be limited. The reverse is true in the U.S., where such a person is considered only American for most purposes.
Both countries have certain laws pertaining to dual nationals. American-German dual nationals may owe taxes in both countries. They must file an annual U.S. income tax return, regardless of whether or not they owe taxes to the U.S. or pay taxes elsewhere. And they must have a valid U.S. passport in order to enter the United States.
An American-German male must register with the U.S. Selective Service System within three months of his eighteenth birthday, and is not necessarily exempt from German military service. None of that, however, affects his dual citizenship.
The rules are sometimes complicated. For more information, check https://de.usembassy.gov/u-s-citizen-services/citizenship-services/, or contact the American embassy, your nearest American consulate, or your local Staatsangehörigkeitsbehörde or Standesamt.
The German Embassy in the USA also has information in English on German citizenship.