You could spend an entire year in Germany....never coming across the Ordnungsamt. Some people could walk around for three or four years before seeing or coming across the Ordnungsamt. In simple terms, they are the local police or municipal police.
Their tasks? Well....parking tickets, littering tickets, dog enforcement, traffic control and investigations related to animal laws (cruelty to animals and such). If you asked a German for a clear definition....the most they will ever offer is parking tickets and they generally laugh when the word 'police' is used to describe these guys.
They report usually to the town mayor, city council or the county commission. Guns? They typically aren't authorized to carry a weapon, but there are exceptions and it's strictly state by state on the rule if they can carry.
Arrests? No. That's the curious thing about their power. They can always issue a citation or a summons. They could perform a German-version of a citizen's arrest....mainly just holding you until the real cops arrive on the scene.
I should add that there are various state and regional titles for the Ordnungsamt....some designed to give a more threatening view of the municipal police (Stadtpolizei for example is the term around Wiesbaden for them).
So your general encounter with these guys.....would likely be parking tickets. There are real cops in the eyes of the local city government, so don't think for a minute of them as a joke. A couple of tickets add up on points, and that is something you really don't need.