die Teilung
division
The process of breaking a whole into two or more parts, sections, or groups.
Die Teilung des Kuchens in acht gleiche Stücke war nicht einfach.
The division of the cake into eight equal pieces was not easy.

division / partition
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division
The process of breaking a whole into two or more parts, sections, or groups.
Die Teilung des Kuchens in acht gleiche Stücke war nicht einfach.
The division of the cake into eight equal pieces was not easy.

The noun 'die Teilung' (division/partition) is derived from the verb 'teilen' (to divide/share). The ending '-ung' turns the verb into a noun that describes the process or result of dividing.
Wir müssen das letzte Stück Pizza noch teilen.
We still have to share the last piece of pizza.
The context makes it clear which meaning of 'Teilung' (division/partition) is meant. In everyday life, it is often about things like food or money. In a historical context, it usually refers to the 'Teilung Deutschlands' (partition of Germany). In biology, people speak of the 'Teilung' (division) of cells.
Die gerechte Teilung der Aufgaben war die größte Herausforderung im Team.
The fair division of the tasks was the biggest challenge in the team.
To describe what is being divided, you usually use the genitive. Alternatively, you can also use 'von' + dative, which often sounds more colloquial.
Die Teilung des Gewinns erfolgte am Ende des Jahres.
The division of the profit took place at the end of the year.
When 'die Teilung' (the partition) is used without further explanation in the context of German history, it almost always refers to the division of Germany into the Federal Republic of Germany and the GDR (1949–1990).
Meine Großeltern haben die Zeit der Teilung noch selbst erlebt.
My grandparents experienced the time of partition themselves.
In biology and medicine, 'Teilung' (division) is a technical term for cell division. The more precise word 'Zellteilung' (cell division) is also often used.
Im Labor beobachten wir die Teilung der Bakterien unter dem Mikroskop.
In the laboratory, we observe the division of the bacteria under the microscope.
separation
'die Trennung' and 'die Teilung' can both mean that something is no longer together as a whole. 'Teilung' is used especially when a whole is divided into parts, for example a country, a room, or a group. 'Trennung' is more general and emphasizes keeping people, things, or areas apart. Learners may confuse the words because both describe something coming apart. When several parts are created from a whole, 'Teilung' is often more precise. When distance or being separate is the main focus, 'Trennung' fits better.

Die Teilung des Raumes in zwei Bereiche machte die Planung einfacher.
Dividing the room into two areas made the planning easier.
Im Labor ist die Trennung der verschiedenen Stoffe sehr wichtig.
In the lab, separating the different substances is very important.