
ordnen
to arrange
To put things into a certain order, system, or place in order to create clarity or structure.
Ich muss dringend die Papiere auf meinem Schreibtisch ordnen.
I urgently need to arrange the papers on my desk.
to arrange / organize
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to arrange
To put things into a certain order, system, or place in order to create clarity or structure.
Ich muss dringend die Papiere auf meinem Schreibtisch ordnen.
I urgently need to arrange the papers on my desk.

to organize
To structure and regulate complex matters or circumstances.
Nach der Scheidung musste sie ihr Leben komplett neu ordnen.
After the divorce, she had to completely reorganize her life.

to settle down
To return to an orderly or clear state after a period of unrest or confusion.
Nach dem ersten Schock begannen sich seine Gedanken langsam zu ordnen.
After the initial shock, his thoughts slowly began to settle down.
The verb 'ordnen' (to arrange / organize) is the action of creating 'Ordnung' (order). When you 'ordnen' something, you put it in order. Both words have the same stem.
Bitte hilf mir, nach der Party wieder Ordnung im Wohnzimmer zu schaffen und alles zu ordnen.
Please help me restore order in the living room after the party and arrange everything.
If you want to state according to which system you arrange something, use the preposition 'nach' (by/according to) with the dative. Typical patterns are arranging something 'nach dem Alphabet' (alphabetically), 'nach Farben' (by color), or 'nach Größe' (by size).
Ich ordne meine Bücher immer nach dem Nachnamen des Autors.
I always arrange my books by the author's last name.
'Ordnen' (to organize) is also used for abstract things, especially when someone structures their life or affairs. Common collocations are 'sein Leben ordnen' (to organize one's life), 'seine Finanzen ordnen' (to get one's finances in order), or 'seine Gedanken ordnen' (to organize one's thoughts).
Nach der Kündigung nahm er sich eine Auszeit, um sein Leben neu zu ordnen.
After being laid off, he took some time off to reorganize his life.
The reflexive form 'sich ordnen' means that something comes into an orderly state by itself. The subject is often something abstract, such as 'die Gedanken' (thoughts) or 'die Situation' (the situation).
Nach dem ersten Schock begannen sich ihre Gedanken langsam wieder zu ordnen.
After the initial shock, her thoughts slowly began to settle down again.
to organize
Organisieren and ordnen can both make documents easier to view and manage. Ordnen places individual documents in their proper place or in a recognizable order more strongly. Organisieren can also refer to the system used to manage the documents over time.

Lea ordnet ihre Unterlagen nach Datum und legt sie in die passenden Ordner.
Lea sorts her documents by date and puts them in the appropriate folders.
Lea organisiert ihre Unterlagen mit einem neuen System aus Ordnern und digitalen Listen.
Lea organizes her documents using a new system of folders and digital lists.