
sich bücken
to bend down
To bend the upper body forward and downward, for example to pick something up from the floor.
Sie musste sich bücken, um den heruntergefallenen Stift aufzuheben.
She had to bend down to pick up the pen that had fallen.
to bend down
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to bend down
To bend the upper body forward and downward, for example to pick something up from the floor.
Sie musste sich bücken, um den heruntergefallenen Stift aufzuheben.
She had to bend down to pick up the pen that had fallen.

to bow down
To submit servilely to another person's will or power; to give up one's principles. This meaning is derogatory.
Er weigerte sich, sich vor der Autorität des Diktators zu bücken.
He refused to bow down before the dictator's authority.
This verb is used almost exclusively reflexively: 'sich bücken' (to bend down). The reflexive pronoun ('mich', 'dich', 'sich', etc.) is a fixed part of it and must not be left out.
Ich bücke mich, um meine Schuhe zu binden.
I bend down to tie my shoes.
When describing the physical movement, the direction or goal is often given with a preposition. Typical constructions are 'sich nach etwas bücken' (to bend down for something) or 'sich zu etwas hinunterbücken' (to bend down toward something).
Der Gärtner bückte sich nach den reifen Erdbeeren.
The gardener bent down for the ripe strawberries.
The figurative meaning 'sich unterwerfen' (to submit) has a strongly negative connotation. It is almost always used with the preposition 'vor' (before/in front of) and a person or power in the dative.
Er war stolz und weigerte sich, sich vor dem Chef zu bücken.
He was proud and refused to bow down before the boss.