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.

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 fulfill
'erfüllen' and 'stillen' can both have to do with wishes or needs. 'Einen Wunsch erfüllen' (to fulfill a wish) means that something desired becomes reality or that someone makes it possible. 'Ein Bedürfnis stillen' (to satisfy a need) means that a strong desire or lack is satisfied, for example hunger, thirst, or curiosity. The words can be confused because both express that something is missing and is then answered in a fitting way. Natural usage is: you use 'erfüllen' with wishes and expectations; you use 'stillen' with hunger, thirst, curiosity, or a need.

Am Wochenende liest sie viel, weil sie ihre Neugier stillen möchte.
On the weekend, she reads a lot because she wants to satisfy her curiosity.
Ich erfülle mir dieses Jahr den Wunsch nach einer Reise ans Meer.
This year I am fulfilling my wish to take a trip to the sea.