brummen
to hum
To produce a deep, monotonous, vibrating sound, typical of machines, large insects, or bears.
Der alte Kühlschrank brummt ziemlich laut.
The old refrigerator is humming pretty loudly.

to hum / grumble / boom
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to hum
To produce a deep, monotonous, vibrating sound, typical of machines, large insects, or bears.
Der alte Kühlschrank brummt ziemlich laut.
The old refrigerator is humming pretty loudly.

'Brummen' (to hum) is an onomatopoeic word. The sound of the word imitates the deep, vibrating noise it describes.
Hörst du auch, wie der Transformator brummt?
Do you hear how the transformer is humming too?
You can almost always recognize the meaning of 'brummen' from the subject of the sentence. Machines, engines, and large insects 'brummen' (hum/buzz) as a sound. People 'brummen' (grumble). Businesses and the economy 'brummen' (are booming/doing well).
Die Klimaanlage brummt leise im Hintergrund.
The air conditioning hums quietly in the background.
When a person 'brummt' (grumbles), this can introduce a dissatisfied statement. Often a subordinate clause with 'dass' (that) follows, or what is said is put in quotation marks.
Er brummte, dass er keine Lust habe, aufzuräumen.
He grumbled that he didn't feel like tidying up.
Saying that a business or the economy 'brummt' (is booming) is colloquial. It means that everything is very successful and a lot of revenue is being generated.
Seit der Eröffnung der neuen U-Bahn-Station brummt unser Café.
Since the opening of the new subway station, our café has been booming.
The expression 'der Kopf brummt' (one's head is pounding) is a fixed phrase for a dull headache. The person who has the pain is often in the dative: 'Mir brummt der Kopf.' (My head is pounding.)
Ich habe zu wenig geschlafen, jetzt brummt mir der Kopf.
I didn't sleep enough; now my head is pounding.
The meaning 'eine Gefängnisstrafe verbüßen' (to serve a prison sentence) is colloquial and belongs to jargon. You can add the length of the sentence.
Er musste für den Diebstahl zwei Jahre brummen.
He had to serve two years for the theft.
to rustle / hiss / sweep
'rauschen' and 'brummen' both describe continuous sounds. 'rauschen' tends to sound brighter, softer, or indistinct, for example with wind, water, leaves, or poor radio reception. 'brummen' is deeper, more even, and often vibrating, for example with an engine or a machine. Learners may confuse the words because both sounds do not seem like clear tones. As a guide: if it sounds deep and heavy, 'brummen' fits better; if it sounds broad, soft, or like background noise, 'rauschen' is more likely to fit.

Zu Hause brummt der Kühlschrank so laut, dass ich ihn im Flur höre.
At home, the refrigerator hums so loudly that I can hear it in the hallway.
Im alten Radio rauscht es, wenn das Signal zu schwach ist.
The old radio hisses when the signal is too weak.