prägen
to shape
To strongly and lastingly influence someone or something in their development, character, or appearance.
Die Zeit im Ausland hat ihn für sein ganzes Leben geprägt.
The time abroad shaped him for his whole life.

to shape / mint
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to shape
To strongly and lastingly influence someone or something in their development, character, or appearance.
Die Zeit im Ausland hat ihn für sein ganzes Leben geprägt.
The time abroad shaped him for his whole life.

The basic meaning of 'prägen' (to mint) is the physical pressing of a pattern into metal, as with a coin. All other meanings are metaphorical: An experience 'prägt' (shapes) a character, you 'prägst' (imprint) a word into your memory, or you 'prägst' (coin) a new term for the language.
Diese Reise hat meine Sicht auf die Welt nachhaltig geprägt.
This trip had a lasting influence on my view of the world.
The object of the verb makes the meaning clear: You prägt Münzen (mint coins, metal), eine Persönlichkeit (shape a personality, person), einen Begriff (coin a term, language), or das Gedächtnis (memory, reflexive: sich etwas einprägen).
Seine Kindheit auf dem Land hat ihn sehr geprägt.
His childhood in the countryside shaped him greatly.
In the meaning of 'beeinflussen' (to influence), 'prägen' (to shape) is often used with abstract nouns such as 'den Charakter' (character), 'das Bild' (the image) of a city, or 'eine Epoche' (an era).
Die Architektur des 19. Jahrhunderts prägt das Stadtzentrum.
The architecture of the 19th century shapes the city center.
'Sich einprägen' (to memorize) is a separable, reflexive verb. The person is in the dative (ich präge MIR ein), and the information is in the accusative. In a main clause, the prefix 'ein' separates.
Ich muss mir deine Telefonnummer gut einprägen.
I have to memorize your phone number well.
In the meaning of 'schaffen' (to create), 'prägen' (to coin) is almost always combined with linguistic elements: 'einen Begriff prägen' (to coin a term), 'ein Wort prägen' (to coin a word), or 'eine Redewendung prägen' (to coin an expression).
Dieser Philosoph hat viele neue Begriffe geprägt.
This philosopher coined many new terms.
to keep
'behalten' and 'sich einprägen' both have to do with remembering. You use 'behalten' when you retain something in your memory, for example a name, a rule, or a number. 'sich einprägen' emphasizes more strongly that something stays very clearly in your memory and that you do not forget it easily. Learners may confuse the words because both say that something is not forgotten. A practical way to think about it is: you actively retain information, but an experience, an image, or a sentence often imprints itself on your memory on its own.

Der Blick vom Berg hat sich mir sofort eingeprägt.
The view from the mountain immediately stuck in my memory.
Ich kann die neue Telefonnummer einfach nicht behalten.
I just can't remember the new phone number.