packen
to pack
To put objects into a suitcase, bag, or box in order to transport them or go on a trip.
Bevor wir in den Urlaub fahren, muss ich noch meinen Koffer packen.
Before we go on vacation, I still have to pack my suitcase.

to pack / grab / manage
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to pack
To put objects into a suitcase, bag, or box in order to transport them or go on a trip.
Bevor wir in den Urlaub fahren, muss ich noch meinen Koffer packen.
Before we go on vacation, I still have to pack my suitcase.

The meaning of 'packen' (to pack / grab / manage) depends heavily on the object. If it refers to an item like 'Koffer' (suitcase) or 'Tasche' (bag), it means 'to put inside'. If it refers to a person, it usually means 'to grab'.
Ich muss noch schnell meine Tasche für die Uni packen.
I still have to quickly pack my bag for university.
When 'packen' (to grab) is used in the sense of 'ergreifen' (to seize/grab), the body part someone is grabbed by often appears with the preposition 'an' and the dative.
Die Mutter packte ihren Sohn an der Hand.
The mother grabbed her son by the hand.
The expression 'es packen' colloquially means 'to manage something' or 'to succeed'. The word 'es' is a fixed part of the idiom here.
Ich lerne viel für die Prüfung und hoffe, dass ich es packe.
I'm studying a lot for the exam and hope that I'll manage it.
In this meaning, the subject of the sentence is often something abstract like a film, a book, or a story. The person who is emotionally moved is in the accusative.
Dieses Lied packt mich jedes Mal wieder.
This song grips me every single time.
The idiom 'seine Sachen packen' means to leave a place, often spontaneously or after a conflict. It is used as a synonym for 'weggehen' (to go away) or 'ausziehen' (to move out).
Wenn du dich nicht an die Regeln hältst, kannst du deine Sachen packen.
If you don't follow the rules, you can pack your things.
to manage / create
'schaffen' and 'packen' can both mean that you successfully manage a task or difficulty. 'schaffen' is neutral for this and fits many situations, for example work, exams, or appointments. 'packen' is more colloquial and often sounds a bit more personal or direct: you are asking whether someone can really manage something. Learners may confuse the words because 'packen' works almost like 'schaffen' in this meaning. In more formal situations, however, 'schaffen' is more natural; among friends, 'packen' often sounds normal.

Keine Sorge, du packst die Prüfung bestimmt.
Don't worry, you'll definitely pass the exam.
Wir schaffen die Präsentation noch vor dem Termin.
We will still manage to finish the presentation before the deadline.