das Geschirr
dishes
A collective term for plates, cups, bowls, and cutlery used for eating and serving food.
Nach dem Abendessen müssen wir noch das Geschirr spülen.
After dinner, we still have to wash the dishes.

dishes / harness
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dishes
A collective term for plates, cups, bowls, and cutlery used for eating and serving food.
Nach dem Abendessen müssen wir noch das Geschirr spülen.
After dinner, we still have to wash the dishes.

The prefix 'Ge-' at the beginning and the ending '-e' at the end often indicate a collective word (collective noun). 'Das Geschirr' (the dishes) is therefore the collection of all plates, cups, and cutlery.
Das ganze Geschirr ist schmutzig.
All the dishes are dirty.
Whether eating dishes or horse harness is meant can be recognized easily from the context. Words like 'spülen' (to wash dishes), 'Küche' (kitchen), or 'Tisch' (table) point to dishes, while 'Pferd' (horse), 'ziehen' (to pull), or 'Wagen' (wagon) refer to animal harness.
Bitte stell das saubere Geschirr in den Schrank.
Please put the clean dishes in the cupboard.
'Geschirr' (dishes) is almost always used in the singular in the sense of eating dishes, even when talking about many items. A plural form ('die Geschirre') is very rare and is used only when referring to different types of dish sets.
Für die Party brauchen wir noch mehr Geschirr.
We need more dishes for the party.
In everyday life, 'Geschirr' (dishes) is often combined with certain verbs. Typical phrases are: 'das Geschirr spülen' (to wash the dishes), or 'abwaschen' (to wash up), 'das Geschirr abtrocknen' (to dry the dishes), and 'den Tisch mit Geschirr decken' (to set the table with dishes).
Wer trocknet heute das Geschirr ab?
Who is drying the dishes today?
service
'Service' and 'Geschirr' can both refer to plates, cups, and similar things used for eating. In this meaning, however, 'Service' is more specific: it means a matching set whose pieces go together, for example for six or twelve people. 'Geschirr' is more general and refers to all plates, cups, bowls, and often also cutlery used in everyday life. The words can be confused because a dinnerware set consists of dishes. If you are simply talking about the things in the kitchen, 'Geschirr' fits. If you mean a matching set, 'Service' is more precise.

Nach dem Abendessen stellt Paul das schmutzige Geschirr in die Spülmaschine.
After dinner, Paul puts the dirty dishes in the dishwasher.
Das alte Service aus Porzellan benutzt Maria nur an Feiertagen.
Maria uses the old porcelain dinnerware set only on holidays.