What is a Bürgeramt?
Bürgeramt is the local citizens office. You can find several offices in your city or town and you can go to any of them to get services that they offer. For example, when you first move to Germany you need to do a city registration (Anmeldung). Or when I renewed my residence permit in Berlin, I had to go to a Bürgeramt.
Generally you need an appointment for a particular service that you need but in some locations, you might not require it.
Here is the entire list of services offered by the Berlin Bürgerämter and all the berlin locations.