
marschieren
to march
To move in a disciplined, rhythmic way, typically in a group such as soldiers or at a demonstration.
Die Soldaten marschierten im Gleichschritt über den Paradeplatz.
The soldiers marched in step across the parade ground.
to march
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to march
To move in a disciplined, rhythmic way, typically in a group such as soldiers or at a demonstration.
Die Soldaten marschierten im Gleichschritt über den Paradeplatz.
The soldiers marched in step across the parade ground.

to march
Colloquial for walking purposefully, quickly, or energetically.
Nach dem Streit ist er wütend aus dem Zimmer marschiert.
After the argument, he marched angrily out of the room.
As a verb of motion, 'marschieren' (to march) always forms the perfect tense with the auxiliary verb 'sein' (to be), not with 'haben' (to have). This applies to all meanings of the word.
Die Wandergruppe ist den ganzen Tag durch den Wald marschiert.
The hiking group marched through the forest all day.
'Marschieren' (to march) is often combined with prepositions that describe a direction or path. Typical prepositions are 'durch' (through), 'zu' (to), 'auf' (onto/toward), and 'in' (into).
Die Demonstranten marschierten durch die Innenstadt.
The demonstrators marched through the city center.
You can tell from the context whether the colloquial meaning is intended. If words like 'Soldaten' (soldiers) or 'Demonstration' (demonstration) are missing and purposeful or emotional walking is described instead, it is the figurative meaning.
Nach dem Telefonat marschierte sie entschlossen in sein Büro.
After the phone call, she marched determinedly into his office.
When 'marschieren' (to march) describes quick, energetic walking, it is used mainly in colloquial language. It often expresses a certain determination or even anger.
Ich muss jetzt los, ich marschiere mal schnell zur Post.
I have to go now; I'm quickly marching over to the post office.
to run
Laufen can colloquially simply mean that someone is going on foot. Marschieren, by contrast, refers to walking in a particularly purposeful, fast, or energetic way. For neutral walking, laufen is suitable; for a determined and distinctive manner of walking, marschieren is more appropriate.

Wir marschieren jetzt entschlossen zum Bahnhof.
We’re marching determinedly to the train station now.
Wir laufen jetzt gemeinsam zum Bahnhof.
We’re walking to the train station together now.