sättigen
to satisfy
To satisfy someone's hunger so that he or she feels full.
Die kräftige Suppe sättigte die ganze Familie.
The hearty soup satisfied the whole family.

to saturate / satisfy
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to satisfy
To satisfy someone's hunger so that he or she feels full.
Die kräftige Suppe sättigte die ganze Familie.
The hearty soup satisfied the whole family.

The verb 'sättigen' (to satisfy/saturate) is closely related to the adjective 'satt' (full). If something 'sättigt' you, you feel 'satt' afterward.
Ein Apfel allein sättigt mich nicht für lange Zeit.
An apple alone doesn't satisfy me for very long.
In the meaning 'Hunger stillen' (to satisfy hunger), the verb is transitive. The food that makes someone full is the subject, and the person who becomes full is the accusative object.
Diese kleine Mahlzeit wird dich kaum sättigen.
This small meal will hardly satisfy you.
In the figurative sense, you can recognize the meaning from objects related to business. Typical combinations are 'den Markt sättigen' (to saturate the market) or 'den Bedarf sättigen' (to satisfy demand).
Irgendwann ist der Markt für diese Produkte gesättigt.
At some point, the market for these products is saturated.
This meaning is limited to a scientific context. You can recognize it from words like 'Lösung' (solution), 'Wasser' (water), or 'Flüssigkeit' (liquid).
Man muss die Lösung mit dem Farbstoff sättigen, um ein gutes Ergebnis zu erzielen.
You have to saturate the solution with the dye to achieve a good result.
When 'sättigen' (to saturate) is used in the visual field, the objects are typically 'die Farben' (the colors), 'ein Foto' (a photo), or 'ein Bild' (an image). The context is usually image editing, photography, or art.
Ich finde das Bild schöner, wenn du die Farben noch etwas sättigst.
I think the image is nicer if you saturate the colors a bit more.
to fulfill
'erfüllen' and 'sättigen' can both mean that a wish, need, or desire is no longer unmet. But 'erfüllen' is broader and fits with wishes, expectations, tasks, or demands. 'sättigen' is used literally for hunger and figuratively when a need is completely met, for example a market or a strong desire. The words can be confused because you can both 'einen Bedarf erfüllen' (fulfill a need) and 'einen Bedarf sättigen' (saturate a need). 'sättigen' sounds stronger here: it is about being completely enough, like being full.

Eine warme Suppe sättigt besser als ein kleiner Snack.
A warm soup is more filling than a small snack.
Mit dem neuen Kurs erfüllt die Schule einen wichtigen Wunsch vieler Eltern.
With the new course, the school fulfills an important wish shared by many parents.