der Körper
body
The entire physical structure of a human or animal, including all organs, bones, and muscles.
Nach dem Sport tut mein ganzer Körper weh.
After exercising, my whole body hurts.

body
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body
The entire physical structure of a human or animal, including all organs, bones, and muscles.
Nach dem Sport tut mein ganzer Körper weh.
After exercising, my whole body hurts.

The noun 'der Körper' (body) is the basis for other important words. The adjective 'körperlich' (physical) describes anything related to the body, and the verb 'verkörpern' (to embody) means to represent or symbolize something.
Körperliche Arbeit kann sehr anstrengend sein.
Physical work can be very exhausting.
Pay attention to the prepositions. German uses 'im Körper' (in the body) for things that are inside (e.g., organs), and 'am Körper' (on the body) for things that are on the surface (e.g., clothing, jewelry).
Das Herz ist ein wichtiges Organ im Körper.
The heart is an important organ in the body.
Common expressions are 'auf seinen Körper achten' (to take care of one's body), 'den Körper fit halten' (to keep the body fit), or 'mit dem ganzen Körper zittern' (to tremble with one's whole body).
Es ist wichtig, auf seinen Körper zu achten und sich gesund zu ernähren.
It is important to take care of your body and eat healthily.
In mathematics or physics, you can recognize this meaning from words like 'geometrisch' (geometric), 'fest' (solid), 'flüssig' (liquid), or 'Himmelskörper' (celestial body). The context here is always scientific or technical.
Wasser ist ein flüssiger Körper.
Water is a liquid body.
When 'Körper' (body) is used in connection with wine, it refers to its taste and fullness. Typical adjectives here are 'voll' (full), 'kräftig' (strong/full-bodied), or 'leicht' (light).
Dieser Rotwein hat einen kräftigen Körper.
This red wine has a strong body.
The meaning 'organized group' is very formal and is used almost only in legal or political texts. You will hardly hear this meaning in everyday life.
Das Parlament ist der gesetzgebende Körper eines Landes.
Parliament is the legislative body of a country.
group
'die Gruppe' and 'der Körper' can, in a special meaning, both refer to several people with a shared function. In normal German, however, 'Gruppe' is the usual and clear word for this. 'Körper' usually means the physical body of a person or animal. As a term for an organized group of people, 'Körper' is rare and very formal, for example in fixed expressions like 'diplomatischer Körper' (diplomatic corps). The words can be confused if 'Körper' is used too generally for a group. In everyday language, you almost always say 'Gruppe'.

Der diplomatische Körper nahm am Montag an der offiziellen Feier teil.
The diplomatic corps took part in the official ceremony on Monday.
Wegen der Prüfung hilft der Lehrer einer Gruppe von Schülern am Morgen.
Because of the exam, the teacher helps a group of students in the morning.