preposition
on, onto
A1
The preposition 'auf' means 'on' or 'onto' (also 'upon'). It is a two-way preposition: use accusative for movement/direction (auf den Tisch) and dative for location (auf dem Tisch). Context decides case. Common in many fixed expressions; governs both accusative and dative accordingly.
Examples
Das Buch liegt auf dem Tisch.
The book is on the table.
Nachdem der Regen aufgehört hatte, setzten sich die Gäste auf die Terrasse, weil sie die frische Luft genießen wollten.
After the rain had stopped, the guests sat down on the terrace because they wanted to enjoy the fresh air.
Stell den Teller auf den Tisch.
Put the plate onto the table.
Details
Mnemonics
Picture a lid 'on' a jar with the lid labeled 'auf' to remember it means 'on/onto'.
rhymes slightly with English 'off' but means 'on' — imagine flipping something 'auf' (open/on).
Notes
A common two-way preposition in German; with movement it often corresponds to 'onto' (accusative), with location to 'on' (dative).