lächeln
to smile
To move the mouth into an expression of joy, friendliness, or amusement without making a loud sound.
Als sie ihn sah, musste sie sofort lächeln.
When she saw him, she immediately had to smile.

to smile
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to smile
To move the mouth into an expression of joy, friendliness, or amusement without making a loud sound.
Als sie ihn sah, musste sie sofort lächeln.
When she saw him, she immediately had to smile.

The word 'lächeln' (to smile) is a diminutive form of 'lachen' (to laugh). So it describes, in a way, a 'small laugh'—quiet and only with the mouth.
Ein Lachen kann man hören, ein Lächeln nur sehen.
A laugh can be heard; a smile can only be seen.
If you want to express that you are smiling directly at a person, the separable verb 'anlächeln' (to smile at) is often used. The object then takes the accusative.
Der Barista lächelte mich freundlich an.
The barista smiled at me in a friendly way.
'Lächeln' (to smile) is often combined with adverbs that describe the type of smile. Common examples are 'freundlich lächeln' (to smile in a friendly way), 'glücklich lächeln' (to smile happily), or 'verlegen lächeln' (to smile shyly/awkwardly).
Als er das Kompliment hörte, lächelte er verlegen.
When he heard the compliment, he smiled shyly.
In the meaning 'to not take something seriously', 'lächeln' (to smile) is always used with the preposition 'über' (about/at) and the accusative. The person smiling often feels superior.
Sie konnte über seine naiven Vorstellungen nur müde lächeln.
She could only smile wearily at his naive ideas.
to laugh
'lachen' and 'lächeln' both show joy or amusement in the face. The most important difference is intensity: with 'lachen' (to laugh), you usually hear a sound; with 'lächeln' (to smile), you only see a friendly or slight facial expression. 'lachen' fits with a joke or a very funny situation. 'lächeln' often fits with greetings, photos, or quiet friendliness. The words can also be confused with 'über': 'über jemanden lachen' (to laugh at someone) can be mean, while 'über etwas lächeln' (to smile at something) often means that you do not take something entirely seriously.

Die Ärztin lächelt freundlich, als sie das Zimmer betritt.
The doctor smiles warmly as she enters the room.
Nach dem Witz musste die ganze Gruppe laut lachen.
After the joke, the whole group had to laugh loudly.