
stillen
to breastfeed
To give an infant the breast so that they drink breast milk.
Die Mutter setzte sich auf eine Bank, um ihr Baby zu stillen.
The mother sat down on a bench to breastfeed her baby.
to breastfeed / satisfy
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to breastfeed
To give an infant the breast so that they drink breast milk.
Die Mutter setzte sich auf eine Bank, um ihr Baby zu stillen.
The mother sat down on a bench to breastfeed her baby.

to satisfy
To satisfy a need, wish, or desire. Often used for hunger, thirst, or curiosity.
Ein großes Glas Wasser stillte seinen Durst.
A large glass of water satisfied his thirst.

to stop / calm
To bring something (e.g. bleeding, pain, an uprising) to an end or to calm it.
Der Sanitäter konnte die Blutung mit einem Druckverband stillen.
The paramedic was able to stop the bleeding with a pressure bandage.
The verb 'stillen' (to breastfeed/satisfy/stop) is closely related to the adjective 'still' (quiet, silent). It describes the action of calming something or someone, or of silencing a need.
Ein Glas Milch am Abend kann den kleinen Hunger stillen.
A glass of milk in the evening can satisfy a small hunger.
You can recognize the exact meaning of 'stillen' (to breastfeed/satisfy/stop) from the object. If it is a baby, it means 'to give the breast'. With needs such as hunger or thirst, it means 'to satisfy'. With problems such as bleeding, it means 'to stop'.
Sie stillt ihr Kind.
She is breastfeeding her child.
In the figurative meaning, 'stillen' (to satisfy) is often used with abstract nouns for needs. Common combinations are 'den Hunger stillen' (to satisfy hunger), 'den Durst stillen' (to quench thirst), and 'die Neugier stillen' (to satisfy curiosity).
Mit einer Tasse Tee konnte sie ihre Sehnsucht nach Wärme stillen.
With a cup of tea, she was able to satisfy her longing for warmth.
to extinguish / delete
Löschen and Durst stillen are both idiomatic. Here, Löschen refers specifically to thirst and presents it figuratively as an urgent condition. Stillen means satisfying a need more generally, so alongside thirst it can also be used with other needs.

Nach dem langen Lauf stillt Mia ihren starken Durst mit Wasser.
After the long run, Mia satisfies her intense thirst with water.
Nach dem langen Lauf löscht Mia ihren starken Durst mit Wasser.
After the long run, Mia quenches her intense thirst with water.